Generic-case complexity of Whitehead's algorithm, revisited

TL;DR

This paper extends previous results on the Whitehead algorithm's efficiency, showing that for a broad class of random elements in free groups, the algorithm performs in quadratic or linear time under certain conditions.

Contribution

It generalizes the understanding of Whitehead's algorithm's generic-case complexity to wider random processes and introduces the notion of $(M, u, heta)$-minimal conjugacy classes.

Findings

Whitehead's algorithm is quadratic on generic random elements in free groups.

For elements close to a filling current, the algorithm runs in linear time.

A wide class of random processes produce elements with quadratic generic-case complexity.

Abstract

In \cite{KSS06} it was shown that with respect to the simple non-backtracking random walk on the free group $F_N=F(a_1,\dots,a_N)$ the Whitehead algorithm has strongly linear time generic-case complexity and that "generic" elements of $F_N$ are "strictly minimal" in their $Out(F_N)$-orbits. Here we generalize these results, with appropriate modifications, to a much wider class of random processes generating elements of $F_N$. We introduce the notion of a ''$(M,\lambda, \epsilon)$-minimal" conjugacy class $[w]$ in $F_N$, where $M\ge 1, \lambda>1$ and $0<\epsilon<1$. Roughly, being $(M,\lambda, \epsilon)$-minimal means that every $\phi\in Out(F_N)$ either increases the length $||w||_A$ by a factor of at least $\lambda$, or distorts the length $||w||_A$ multiplicatively by a factor $\epsilon$-close to $1$, and that the number of automorphically minimal $[u]$ in the orbit $Out(F_N)[w]$ is bounded by $M$. We then show that if a conjugacy class $[w]$ in $F_N$ is sufficiently close to a "filling" projective geodesic current $[\nu]\in PCurr(F_N)$, then, after applying a single "reducing" automorphism $\psi=\psi(\nu)\in Out(F_N)$ depending on $\nu$ only, the element $\psi([w])$ is $(M,\lambda, \epsilon)$-minimal for some uniform constants $M,\lambda,\epsilon$. Consequently, for such $[w]$, Whitehead's algorithm for the automorphic equivalence problem in $F_N$ works in quadratic time on the input $([w], [w'])$ where $[w']$ is arbitrary, and in linear time if $[w']$ is also projectively close to $[\nu]$. We then show that a wide class of random processes produce "random" conjugacy classes $[w_n]$ that projectively converge to some filling current in $PCurr(F_N)$. For such $[w_n]$ Whitehead's algorithm has at most quadratic generic-case complexity.

Equations

$$||\phi {\nu }_A|{|}_A=\frac{||\phi ({\nu }_A)|{|}_A}{||{\nu }_A|{|}_A}\ge {\lambda }_0=1+\frac{2N-3}{2N^2-N}>1$$

$$\tau (x)\in \{x,xa,a^{-1}x,a^{-1}xa\}.$$

$$||{\tau }_i\dots {\tau }_1(w)|{|}_A=n.$$

$$S_{i+1}=S_i\cup \{\tau ([u])|[u]\in S_i,\tau \in {\mathcal{W}}_N\text{ and }||\tau (u)|{|}_A=n\}.$$

$$||\phi (u)|{|}_A\ge ||v_j|{|}_A\ge {\lambda }^{\prime }(1-{\epsilon }^{\prime })||u|{|}_A\ge \lambda ||u|{|}_A.$$

$$||u|{|}_A>||{\tau }_1(u)|{|}_A>||{\tau }_2{\tau }_1(u)|{|}_A>\cdots >||{\tau }_k\dots {\tau }_2{\tau }_1(u)|{|}_A$$

$$\mathrm{rank}Sta{b}_{\mathrm{Out}(F_N)}([u])\le K(N,M).$$

$$||u_i|{|}_A\le (1+\epsilon )||u|{|}_A$$

$${\eta }_g:=\sum _{h\in F_N/⟨g⟩}{\delta }_{h(g^{-\infty },g^{\infty })}+{\delta }_{h(g^{\infty },g^{-\infty })}$$

$$\underset{n\to \infty }{\lim }⟨v,{\eta }_n{⟩}_{\Gamma }=⟨v,\eta {⟩}_{\Gamma }.$$

$$⟨v,\eta {⟩}_{\Gamma }=\sum _{e\in E\Gamma \text{ with }ve\in {\Omega }_{k+1}(\Gamma )}⟨ve,\eta {⟩}_{\Gamma }=\sum _{e^{\prime }\in E\Gamma \text{ with }{e}^{\prime }v\in {\Omega }_{k+1}(\Gamma )}⟨e^{\prime }v,\eta {⟩}_{\Gamma }.$$

$$\text{mbox}Supp(\eta ):={\partial }^2{F}_N-\cup \{U\subseteq {\partial }^2{F}_N|U\text{ is open and }\eta (U)=0\}.$$

$$⟨-,-⟩:{\overline{\text{mbox}cv}}_N\times \text{mbox}Curr(F_N)\to {\mathbb{R}}_{\ge 0}$$

$$⟨T,\phi \eta ⟩=⟨T\phi ,\eta ⟩.$$

$$0=⟨T,{\eta }_{a_i}⟩=||a_i|{|}_T$$

$$0=⟨T,{\eta }_{a_i{a}_j}⟩=||a_i{a}_j|{|}_T$$

$$D_A(\nu ):=\{⟨T_A,\phi \nu ⟩|\phi \in \mathrm{Out}(F_N)\}$$

$$R_{\Im }(\nu )=\{\psi \in \mathrm{Out}(F_N)|\psi {\nu }^{\prime }\in \Im \}.$$

$$R_{\Im }[u]=S.$$

$${\lambda }_A(\nu )>{\lambda }_1>{\lambda }_1(1-\epsilon )>\lambda >1+\epsilon .$$

$$1-\epsilon \le \frac{||\psi \eta |{|}_A}{||\eta |{|}_A}\le 1+\epsilon .$$

$$\frac{||\tau {\nu }^{\prime }|{|}_A}{||{\nu }^{\prime }|{|}_A}\ge {\lambda }_1$$

$$\left|\frac{⟨v,\eta ⟩}{||\eta |{|}_A}-\frac{⟨v,\nu ⟩}{||\nu |{|}_A}\right|\le {\epsilon }_0.$$

$$\left|\frac{⟨v,w⟩}{||w|{|}_A}-\frac{⟨v,\nu ⟩}{||\nu |{|}_A}\right|\le {\epsilon }_0$$

$$\underset{n\to \infty }{\lim }[{\eta }_{w_n}]=[\nu ]$$

$$\lim_{n\to\infty}Pr\big{(}\mathfrak{W}[W_{n}]\text{ is $(M,\lambda,\varepsilon)$-minimizing}\big{)}=1.$$

$$\lim_{n\to\infty}Pr\big{(}\mathfrak{W}[\eta_{W_{n}}]\in U_{0}\big{)}=1.$$

$$Pr(\zeta \in \Omega |\zeta \text{ satisfies }\mathcal{E})=1.$$

$$\mathcal{W}=W_1,W_2,\dots ,W_n,\dots$$

$$p_{\mathcal{X}}^{(n)}(s,s^{\prime })=\sum _{s^{\prime \prime }\in S}{p}_{\mathcal{X}}^{(n-1)}(s,s^{\prime \prime })p_{\mathcal{X}}(s^{\prime \prime },s^{\prime }).$$

Full text

The author was supported by the individual NSF grants DMS-1710868 and DMS-1905641

1. 1 Introduction

2. 2 Whitehead’s algorithm

3. 3 $(M,\lambda,\varepsilon)$-minimality and Whitehead’s algorithm

4. 3.1 Main definitions

5. 3.2 Behavior of Whitehead’s algorithm

6. 3.3 Algorithmic detectability

7. 4 Geodesic currents on free groups

8. 4.1 Basic notions

9. 4.2 Simplicial charts and weights

10. 4.3 Geometric intersection form

11. 5 Filling geodesic currents

12. 6 Filling currents and $(M,\lambda,\varepsilon)$-minimality

13. 7 Group random walks as a source of $(M,\lambda,\varepsilon)$minimality

14. 8 Finite-state Markov chains and the frequency measures

15. 8.1 Finite-state Markov chains.

16. 8.2 Iterated Markov Chains

17. 8.3 Quasi-inversions

18. 9 Graph-based non-backtracking random walks

1. Introduction

Let $F_{N}=F(A)$ be a free group of finite rank $N\geq 2$, with a fixed free basis $A=\{a_{1},\dots,a_{N}\}$. The automorphism problem for $F_{N}$ asks, given two freely reduced words $w,w^{\prime}\in F_{N}=F(A)$, whether there there exists $\varphi\in{\rm Aut}(F_{N})$ such that $w^{\prime}=\varphi(w)$, that is, whether ${\rm Aut}(F_{N})w={\rm Aut}(F_{N})w^{\prime}$. A complete algorithmic solution to this problem was provided in 1936 classic paper of Whitehead [46], via the procedure that came to be called Whitehead’s algorithm. We briefly recall how this algorithm works, and refer the reader to Section 2 below. For an element $g\in F_{N}$, we denote by $|g|_{A}$ and by $||g||_{A}$ the freely reduced length and the cyclically reduced length of $g$ with respect to $A$ accordingly. We also denote by $[g]$ the conjugacy class of $g$ in $F_{N}$. For $w,w^{\prime}\in F_{N}$ we have ${\rm Aut}(F_{N})w={\rm Aut}(F_{N})w^{\prime}$ if and only if ${\rm Out}(F_{N})[w]={\rm Out}(F_{N})[w^{\prime}]$. For that reason we usually think of the automorphism problem in $F_{N}$ in this latter form, as the question about $[w],[w^{\prime}]$ being in the same ${\rm Out}(F_{N})$-orbit. We denote $\mathcal{C}_{N}=\{[g]|g\in F_{N}\}$. The group ${\rm Aut}(F_{N})$ has a particularly nice finite generating set $\mathcal{W}_{N}$ of so-called Whitehead automorphisms or Whitehead moves (we use the same terminology for the images of Whitehead automorphisms in ${\rm Out}(F_{N})$); see Definition 2.1 below. Whitehead moves are divided into two types: Whitehead moves $\tau$ of the first kind have the form $a_{i}\mapsto a_{\sigma(i)}^{\pm 1}$ for some permutation $\sigma\in S_{n}$. They have the property that for each $w\in F_{N}$ $||\tau(w)||_{A}=||w||_{A}$. Whitehead moves of the second kind can change the cyclically reduced length of an element of $F_{N}$.

An element $[g]\in\mathcal{C}_{N}$ is called ${\rm Out}(F_{N})$-minimal if for every $\varphi\in{\rm Out}(F_{N})$ we have $||g||_{A}\leq||\varphi(g)||_{A}$. For $[g]\in\mathcal{C}_{N}$ we denote by $\mathcal{M}([g])$ the set of all ${\rm Out}(F_{N})$-minimal elements in the orbit ${\rm Out}(F_{N})[g]$.

An element $[g]\in\mathcal{C}_{N}$ is called Whitehead-minimal if for every Whitehead move $\tau\in\mathcal{W}_{N}$ we have $||g||_{A}\leq||\tau(g)||_{A}$. Whitehead’s “peak reduction lemma” implies the following two key facts: If $[w]$ is not ${\rm Out}(F_{N})$-minimal, then there exists $\tau\in\mathcal{W}_{N}$ such that $||\tau(w)||_{A}<||w||_{A}$. This fact already has the following important implication: an element $[w]\in\mathcal{C}_{N}$ is ${\rm Out}(F_{N})$-minimal if and only if $[w]$ is Whitehead minimal.

The second fact says that for two ${\rm Out}(F_{N})$-minimal $[u],[u^{\prime}]\in\mathcal{C}_{N}$ we have ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$ if and only if $||w||_{A}=||w^{\prime}||_{A}$ and there exists a finite length-stable chain of Whitehead moves $\tau_{1}\dots,\tau_{k}\in\mathcal{W}_{N}$ (with $k\geq 0$) such that $\tau_{k}\dots\tau_{1}[u]=[u^{\prime}]$ and that $||\tau_{i}\dots\tau_{1}[u]||_{A}=||u||_{A}$ for all $i\leq k$. Whitehead’s algorithm on the input $([w],[w^{\prime}])$ consists of two parts. The first one, the Whitehead minimization algorithm, starting from $[w]\in\mathcal{C}_{N}$ consists of iteratively looking for a Whitehead move that decreases the cyclically reduced length of an element. Once we have arrived at $[u]$ where no such moves are available, we know that $[u]$ is a Whitehead-minimal and hence ${\rm Out}(F_{N})$-minimal element of the orbit ${\rm Out}(F_{N})[w]$. Since the set $\mathcal{W}_{N}$ is finite and fixed, this process runs in at most quadratic time in terms of $||w||_{A}$. Also, do the same thing to $[w^{\prime}]$ to produce an ${\rm Out}(F_{N})$-minimal element $[u^{\prime}]$ of the orbit ${\rm Out}(F_{N})[w]$. If $||u||_{A}\neq||u^{\prime}||_{A}$, then ${\rm Out}(F_{N})[w]\neq{\rm Out}(F_{N})[w^{\prime}]$ and we are done. The second, hard, part of Whitehead’s algorithm, that we call Whitehead’s stabilization algorithm, deals with the case where $||u||_{A}=||u^{\prime}||_{A}=n\geq 1$. In this case one looks for a length-stable chain $\tau_{1}\dots,\tau_{k}\in\mathcal{W}_{N}$ of Whitehead’s move which satisfies $\tau_{k}\dots\tau_{1}[u]=[u^{\prime}]$. Since the ball of radius $n$ in $F_{N}(A)$ has exponential size in $n$, this second process has a priori exponential in $n$ time complexity. Although a few incremental improvements have been obtained over the years (e.g. see [10, 31, 35, 36, 40, 41, 44]), the questions about the computational complexity of the automorphism problem in $F_{N}$ and about the actual worst-case complexity of Whitehead’s algorithm remain wide open and the exponential time bound is the best one known in general. The only exception is the case of rank $N=2$ where it is known that Whitehead’s algorithm works in polynomial (in fact, quadratic) time [41, 31]. For the general case $N\geq 2$, the best known partial results are due to Donghi Lee [35, 36], who proved that Whitehead’s algorithm terminates on $w\in F_{N}$ in polynomial time (with degree of the polynomial depending on $N$), if some ${\rm Out}(F_{N})$-minimal element $[u]\in\mathcal{M}([w])$ satisfies a certain technical condition.

In [29] Kapovich, Schupp and Shpilrain initiated a probabilistic study of Whitehead’s algorithm, that is, its behavior on “random” or “generic” inputs in $F_{N}$. In that paper “generic” meant for a large $n\geq 1$, either choosing a uniformly at random freely reduced word of length $n$ in $F(A)$, or taking a a uniformly at random cyclically reduced word of length $n$ in $F(A)$. It turned out that on such “generic” input both parts of Whitehead’s algorithm work very fast. As defined in [29], an element $[w]\in\mathcal{C}_{N}$ is called strictly minimal if for every non-inner $\tau\in\mathcal{W}_{N}$ of the second kind we have $||w||_{A}<||\tau(w)||_{A}$. Thus, in particular, a strictly minimal element is Whitehead-minimal and therefore ${\rm Out}(F_{N})$-minimal. Thus the Whitehead minimization algorithm on $[w]$ terminates in a single step with $[u]=[w]$, and takes linear time in $||w||_{A}$. Also, in this case (for $[w]$ strictly minimal), if $[u^{\prime}]$ is another ${\rm Out}(F_{N})$-minimal element with $||u^{\prime}||_{A}=||u||_{A}=n$ and with ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$ any length-stable chain $\tau_{1},\dots,\tau_{k}$ connecting $[w]=[u]$ to $[u^{\prime}]$ consists only of inner automorphisms and Whitehead moves of the first kind. By composing them we see that ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$ if and only if there exists $\tau\in\mathcal{W}_{N}$ of the second kind such that $\tau[u]=[u^{\prime}]$. Thus in this case the Whitehead stabilization algorithm also terminates in linear time in $n=||w||_{A}$. The overall complexity of Whitehead’s algorithm on the input $[w],[w^{\prime}]$, where $[w]$ is strictly minimal, is $O(\max\{||w||_{A},||w^{\prime}||_{A}^{2}\})$. A key probabilistic result of [29] says that a “generic” (in the above basic sense of taking a uniformly random freely reduced or cyclically reduced word of length $n$) element $[w]\in\mathcal{C}_{N}$ is strictly minimal. Therefore, if both $[w],[w^{\prime}]$ are “generic” in this sense, Whitehead’s algorithm on input $[w],[w^{\prime}]$ runs in $O(\max\{||w||_{A},||w^{\prime}||_{A}\})$ time; and if $[w]$ is generic and $[w^{\prime}]$ is arbitrary, it runs in $O(\max\{||w||_{A},||w^{\prime}||_{A}^{2}\})$ time. The results of [29] were generalized in [44] for the version of Whitehead’s algorithm for ${\rm Out}(F_{N})$-orbits of conjugacy classes of finitely generated subgroups of $F_{N}$.

The proof in [29] that “generic” $[w]$ in $F_{N}$ is strictly minimal crucially relied on the fact that for such $[w]$ the weights (normalized by $||w||_{A}$) on edges in the Whitehead graph of $w$ are close to being uniform. Roughly, that means that frequencies of 1-letter and 2-letter subwords in $[w]$ are close to being uniform (e.g. that for $i=1,\dots,N$ the frequency of each $a_{i}^{\pm 1}$ in $[w]$ is close to $\frac{1}{N}$). This close-to-uniform property of frequencies no longer holds if $[w]$ generated by other random processes.

Example 1.1*.*

For example, consider the case $N=2$ and $F_{2}=F(a,b)$. Let $w_{n}$ be a positive word in $\{a,b\}$ of length $n$, where every letter is chosen independently, with probability $p(a)=1/10$ and $p(b)=9/10$. Then the frequency of $a$ in a ”random” $w_{n}$ will tend to $1/10$ as $n\to\infty$. Moreover, it is not hard to see that $w_{n}$ will not be strictly minimal. Here is an informal argument. In this case $w_{n}$ will contain $\frac{81}{1000}n+o(n)$ occurrences of $ab^{2}$, as well as $\frac{1}{1000}n+o(n)$ occurrences of $a^{3}$ and $\frac{9}{1000}n+o(n)$ occurrences of $aba$. Consider the Whitehead move $\tau(a)=ab^{-1},\tau(b)=b$. Note that $\tau(ab^{2})=ab$. The portion of $w_{n}$ covered by the $\frac{81}{1000}n+o(n)$ occurrences of $ab^{2}$ has total length $\frac{243}{1000}n+o(n)$ but its image under $\tau$ has total length $\frac{162}{1000}n+o(n)$. Since $\tau(aba)=aab^{-1}$, the image of the portion of $w_{n}$ covered by occurrences of $aba$ in $w_{n}$ does not change in length in $\tau(w_{n})$. The portion of $w_{n}$ covered by the $\frac{1}{1000}n+o(n)$ occurrences of $a^{3}$ has total length $\frac{3}{1000}n+o(n)$, and its image in $\tau(w_{n})$ has total length $\frac{6}{1000}n+o(n)$ there. One can conclude from here that $||w_{n}||_{A}-||\tau(w_{n})||_{A}\geq\frac{78}{1000}n+o(n)$. Thus $||\tau(w_{n})||_{A}<||w_{n}||_{A}$ and, moreover (since $||w_{n}||_{A}=n$), $\frac{||\tau(w_{n})||_{A}}{||w_{n}||_{A}}\leq\frac{922}{1000}+o(1)$. Hence $w_{n}$ badly fails to be strictly minimal.

In the present paper we consider the generic-case behavior of Whitehead’s algorithm on random inputs for much more general types of random processes than in the [29] setting (in particular, including Example 1.1). Our results have some similarities to the results from [29] but, of course, with important differences that are inherently necessary, as demonstrated by Example 1.1.

The main notions replacing strict minimality are the notions of an $(M,\lambda,\varepsilon)$-minimal element $[u]\in\mathcal{C}_{N}$ and of $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal element $[u]\in\mathcal{C}_{N}$ (where $M\geq 1$, $\lambda>1$ and $0<\varepsilon<1$); see Definition 3.1 and Definition 3.3 below. Roughly, $[u]$ being $(M,\lambda,\varepsilon)$-minimal means that $[u]$ belongs to a subset $S\subseteq{\rm Out}(F_{N})[u]$ of cardinality at most $M$ such that for each element $[u^{\prime}]$ of $S$ an arbitrary $\varphi\in{\rm Out}(F_{N})$ either increases the length $||u^{\prime}||_{A}$ by a factor of at least $\lambda$, or distorts the length multiplicatively by a factor $\varepsilon$-close to $1$. For $[u]$ being $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal the definition is similar, but instead of arbitrary $\varphi\in{\rm Out}(F_{N})$ we only require these conditions to hold for arbitrary $\tau\in\mathcal{W}_{N}$. From the definitions we see that being $(M,\lambda,\varepsilon)$-minimal directly implies being $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal. The converse is almost true: given $\lambda,\varepsilon$, for all ”sufficiently stringent” $\lambda^{\prime}>\lambda$ and $0<\varepsilon^{\prime}<\varepsilon$ being $(M,\lambda^{\prime},\varepsilon^{\prime},\mathcal{W}_{N})$-minimal implies being $(M,\lambda,\varepsilon)$-minimal. See Proposition 3.5 below for a precise statement. For fixed $M,\lambda,\varepsilon$, deciding if an element $[u]\in\mathcal{C}_{N}$ is $(M,\lambda^{\prime},\varepsilon^{\prime},\mathcal{W}_{N})$-minimal can be done in linear time in $||u||_{A}$, while the algorithm for deciding if $[u]$ is $(M,\lambda,\varepsilon)$-minimal has a priori exponential time complexity. See Section 3.3 below for details.

We summarize the main results of the present paper:

$\bullet$ We show in Theorem 3.11 that Whitehead’s algorithm on an input $[w],[w^{\prime}]$ works fast if at least one of the inputs is $(M,\lambda,\varepsilon)$-minimal: Whitehead minimization algorithm works in linear time on any $(M,\lambda,\varepsilon)$-minimal element $[w]$; if both $[w],[w^{\prime}]$ are $(M,\lambda,\varepsilon)$-minimal, the full Whitehead algorithm the input $([w],[w^{\prime}])$ works in linear time; if $[w]$ is $(M,\lambda,\varepsilon)$-minimal and $[w^{\prime}]$ is arbitrary, the full Whitehead algorithm the input $([w],[w^{\prime}])$ works in time $O(||w||_{A},||w^{\prime}||_{A}^{2})$. Also, for an $(M,\lambda,\varepsilon)$-minimal $[w]$ the stabilizer $Stab_{{\rm Out}(F_{N})}([w])$ has uniformly bounded rank rank (the smallest cardinality of a generating set).

$\bullet$ We exhibit a rich source of $(M,\lambda,\varepsilon)$-minimal elements. We prove that if $\nu\in\mbox{Curr}(F_{N})$ is a “filling” geodesic current then there exist $M\geq 1,\lambda>1,\varepsilon>0$, a neighborhood $U$ of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$, and a finite set $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ of $\leq M$ “shortening” automorphisms with the following property: For every $[w]$ which belongs in $U$ (when $[w]$ is viewed as a projective current), and every $\psi\in\mathfrak{W}$, the element $[\psi(w)]$ is $(M,\lambda,\varepsilon)$-minimal, and, moreover $\mathcal{M}([w])\subseteq\mathfrak{W}([w])$. We also produce several sources of “filling” currents.

$\bullet$ We define the notion of an $F_{N}$-valued random process $\mathcal{W}=W_{1},W_{2},\dots$ being adapted to a current $0\neq\nu\in\mbox{Curr}(F_{N})$. We prove that if $\mathcal{W}$ is adapted to a filling current $\nu$ then Whitehead’s algorithm has low complexity when one or both of $[w],[w^{\prime}]$ are “randomly” generated by $\mathcal{W}$. These conclusions, obtained in Theorem 6.11 and Theorem 6.12, are the main genericity results of this paper.

$\bullet$ We show, in Theorem 7.6, that for a large class of “group random walks” $\mathcal{W}=W_{1},W_{2},\dots$ on $F_{N}$, the walk is adapted to some filling current $[\nu]$ and hence Theorem 6.11 and Theorem 6.12 apply.

$\bullet$ We also show that for a large class of “graph non-backtracking random walks” $\mathcal{W}=W_{1},W_{2},\dots$ on $F_{N}$, the walk is adapted to some filling current $[\nu]$ (see Theorem 9.11, Theorem 9.12 and Proposition 9.14) and hence again Theorem 6.11 and Theorem 6.12 apply.

In [29] the probabilistic results about Whitehead’s algorithm are stated in terms of generic-case complexity. This notion, introduced in [28], is designed to capture practically observable (as distinct from worst-case and average-case) behavior of various algorithms. In [28] generic-case complexity in the $F_{N}=F(A)$ context is defined via asymptotic density, that is, essentially, the uniform probability measure on large spheres or balls in $F(A)$; the same definition is still used in [29]. Since then the notion of generic-case complexity has been significantly expanded and generalized, to allow for more general and more natural models of random generation of inputs; see [42] for some background and further details. All the probabilistic complexity results about Whitehead’s algorithm obtained in this paper are, in fact, generic-case complexity results. However, to be precise, we state all these results exactly, precisely and explicitly (including quantification of various constants) in terms of the random processes involved, rather than using the language of generic-case complexity. Note that the case of a simple non-backtracking random walk on $F(A)$, which was the context of the results in [29], is a very special case of the random process $\breve{\mathcal{W}}$ considered Theorem 9.12. We expect the results of Section 3 about $(M,\lambda,\varepsilon)$-minimal elements to be of independent interest, apart from any probabilistic applications.

Geodesic currents provide a measure-theoretic generalization of the notion of a conjugacy class. Geodesic currents, originally introduced by Bonahon [5] in the context of hyperbolic surfaces, proved particularly useful in recent years in the study of ${\rm Out}(F_{N})$ and of the Culler-Vogtmann Outer space, see e.g. [11, 17, 4, 20]. A key tool in the theory is the notion of a geometric intersection form between currents and points of the Thurston-like closure of the Outer space. The intersection form was developed by Kapovich and Lustig [26, 27]. The connection between currents and generic-case complexity was first pointed out in our article [24], but this connection is explored in detail for the first time in the present paper. In particular, the intersection form defines, for every $\nu\in\mbox{Curr}(F_{N})$, the “length” $||\nu||_{A}\geq 0$ of $\nu$ with respect to $A$. For $1\neq w\in F_{N}$, we have $||\eta_{w}||_{A}=||w||_{A}$, where $\eta_{w}\in\mbox{Curr}(F_{N})$ is the “counting” current associated with $[w]$.

Example 1.2* (Simple non-backtracking random walk on $F_{N}$).*

Consider the simple nonbacktracking random walk $\mathcal{W}=W_{1},W_{2},\dots,W_{n},\dots$ of $F_{N}$ with respect to $A$, as is done in [29]. This means that the $W_{n}=X_{1}\dots X_{n}$ is a freely reduced word of length $n$ in $F(A)$, where the first letter $X_{1}$ is chosen uniformly at random from $A^{\pm 1}$ with probability $\frac{1}{2N}$ each; and if the $i$-th letter $X_{i}=a\in A^{\pm 1}$ is already chosen, the letter $X_{i+1}$ is chosen uniformly at random from $A^{\pm 1}-\{a\}$, with probability $\frac{1}{2N-1}$ for each element there. Thus $W_{n}$ induces the uniform probability distribution on the $n$-sphere in $F(A)$, where every element of the sphere has probability $\frac{1}{2N(2N-1)^{n-1}}$. We will explain here the properties of the walk $\mathcal{W}$ in the terminology of this paper, omitting the detailed justification of these properties.

For a.e. trajectory $w_{1},w_{2},\dots$ of the walk $\mathcal{W}$ we have $\lim_{n\to\infty}\frac{1}{n}\eta_{w_{n}}=\nu_{A}$ in $\mbox{Curr}(F_{N})$, where $\nu_{A}$ is the uniform current on $F_{N}$ corresponding to $A$ (see Definition 4.7). Thus, in the language of the present paper, the walk $\mathcal{W}$ is adapted to $\nu_{A}$. Moreover, $\nu_{A}$ has full support in $\partial^{2}F_{N}$ and therefore $\nu_{A}$ is filling, by Proposition 5.2. Also, we have $||\nu_{A}||_{A}=1$. Let $\Im\subseteq{\rm Out}(F_{N})$ be the set of all Whitehead automorphisms of the first kind. Put $\mathfrak{W}=\Im\subseteq{\rm Out}(F_{N})$. The results of [24, 22] imply that for any $\varphi\in{\rm Out}(F_{N})$, we have $\varphi(\nu_{A})=\nu_{A}$ if $\varphi\in\Im$, and

[TABLE]

if $\varphi\not\in\Im$. This fact implies that if we choose and fix any $1<\lambda<\lambda_{0}$, then any $[w]$ that is sufficiently close to $[\nu_{A}]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ is strictly minimal (in particular $||\varphi(w)||_{A}=||w||_{A}$ for every $\varphi\in\Im$), and, moreover, for any $\varphi\not\in\Im$ we have $||\varphi([w])||_{A}/||w||_{A}\geq\lambda$. Then for any sufficiently small $\varepsilon>0$, for $n\to\infty$ our “random” $w_{n}$ is $(M,\lambda,\varepsilon)$-minimal with $M=\#\Im$, and the set $S_{n}=\Im([w_{n}])=\mathfrak{W}([w_{n}])$ is $(M,\lambda,\varepsilon)$-minimazing (in the sense of Definition 3.1). This example is the simplest case illustrating how our definitions and results work.

Remark 1.3* (A note on the speed of convergence).*

In [28, 29] the main results are stated in terms of “strong genericity”, meaning that various probabilities converging to $1$ do so exponentially fast as $n\to\infty$. Parts (b) of Theorem 6.11 and Theorem 6.12 are also stated in terms of probabilities of various events at step $n$ converging to $1$ as $n\to\infty$. We do not include the speed of convergence estimates there because for the moment our main new ”group random walks” application, Theorem 7.6, does not come with a speed of convergence estimate. The reason is that the proof of this theorem relies on the use of a recent result of Gekhtman [19, Theorem 1.5] about approximating harmonic measure by counting currents along a random walk on a word-hyperbolic group acting on a $CAT(-1)$ space does not have any speed of convergence estimates. We expect that Gekhtman’s result actually holds in much greater generality (e.g. for an arbitrary geometric action of a nonelementary word-hyperbolic group $G$, and with much milder assumptions on the measure $\mu$ defining the walk), with exponentially fast convergence. Once that is proved, the applications of Theorem 6.11 and Theorem 6.12 to the group random walk context can be supplied with the speed of convergence estimates. (Definition 6.9 of a random process adapted to a current would have to be refined to include quantificantion by the speed of convergence.) On the other hand, in the context of our results about graph-based non-backtracking random walks, namely Theorem 9.11, Theorem 9.12, one can already show that the convergence is either exponentially or slightly subexponentially fast.

We are extremely grateful to Vadim Kaimanovich and Joseph Maher for many helpful discussions about random walks, for help with the references and for clarifying some random walks arguments. In particular the proof of Proposition 7.5 was explained to us by Kaimanovich. We are also grateful to the organizers of the March 2019 Dagstuhl conference ”Algorithmic Problems in Group Theory” for providing impetus and motivation for completing this paper.

2. Whitehead’s algorithm

Our main background reference for Whitehead’s algorithm Lyndon and Schupp, Chapter I.4 [37], and we refer the reader there for additional details. Some other useful details and complexity results are available in [29, 44]. We recall the basic definitions and results here.

In this section we fix a free group $F_{N}=F(A)$ of rank $N\geq 2$, with a fixed free basis $A=\{a_{1},\dots,a_{N}\}$. Put $\Sigma_{A}=A\sqcup A^{-1}$. We will also denote by $\mathcal{C}_{N}$ the set of all $F_{N}$-conjugacy classes $[g]$ where $g\in F_{N}$.

Definition 2.1** (Whitehead automorphisms).**

A Whitehead automorphism of $F_{N}$ with respect to $A$ is an automorphism $\tau\in{\rm Aut}(F_{N})$ of $F_{N}$ of one of the following two types:

(1) There is a permutation $t$ of $\Sigma_{A}$ such that $\tau|_{\Sigma_{A}}=t$. In this case $\tau$ is called a relabeling automorphism or a Whitehead automorphism of the first kind.

(2) There is an element $a\in\Sigma_{A}$, the multiplier, such that for any $x\in\Sigma_{A}$

[TABLE]

In this case we say that $\tau$ is a Whitehead automorphism of the second kind. (Note that since $\tau$ is an automorphism of $F_{N}$, we always have $\tau(a)=a$ in this case).

We also refer to the images of Whitehead automorphisms in ${\rm Out}(F_{N})$ as Whitehead moves and sometimes again as Whitehead automorphisms. We denote by $\mathcal{W}_{N}$ the set of all Whitehead moves $\tau\in{\rm Out}(F_{N})$ such that $\tau\neq 1$ in ${\rm Out}(F_{N})$.

Note that for any $a\in\Sigma_{A}$ the inner automorphism $ad(a)\in{\rm Aut}(F_{N})$ is a Whitehead automorphism of the second kind. Note also that if $\tau\in\mathcal{W}_{N}$ then $\tau^{-1}\in\mathcal{W}_{N}$.

To simplify the exposition, we formulate all the definitions and results related to Whitehead’s algorithm in terms of conjugacy classes of elements of $F_{N}$. In this context we usually think of an input $[w]\in\mathcal{C}_{N}$ as given by a cyclically reduced word $w\in F(A)$ and the complexity of various algorithms is estimated in terms of $||w||_{A}$. Since for $w\in F_{N}$ we have $||w||_{A}\leq|w|_{A}$, and since it takes linear time in $|w|_{A}$ to find a cyclically reduced form of $w\in F(A)$ (see [29] for additional discussion on this topic), the same complexity estimates hold in terms of $|w|_{A}$.

Definition 2.2** (Minimal and Whitehead-minimal elements).**

A conjugacy class $[w]\in\mathcal{C}_{N}$ is $Out(F_{N})$-minimal with respect to $A$ if for every $\varphi\in{\rm Out}(F_{N})$ we have $||w||_{A}\leq||\varphi(w)||_{A}$.

A conjugacy class $[w]\in\mathcal{C}_{N}$ is Whitehead-minimal with respect to $A$ if for every Whitehead move $\tau\in\mathcal{W}_{N}$ we have $||w||_{A}\leq||\tau(w)||_{A}$.

For $[w]\in\mathcal{C}_{N}$, denote \mathcal{M}([w])=\{[u]\in{\rm Out}(F_{N})[w]|[u]\text{ is Out(F_{N})-minimal}\}.

Note that, by definition, an $Out(F_{N})$-minimal $[w]$ is necessarily Whitehead-minimal.

Definition 2.3** (Automorphism graph).**

The automorphism graph of $F_{N}$ is an oriented labelled graph $\mathcal{T}$ defined as follows.

The vertex set $V\mathcal{T}$ is $\mathcal{C}_{N}$, the set of all conjugacy classes $[w]$ where $w\in F_{N}$. The edges of $\mathcal{T}$ are defined as follows. Suppose that $[w]\neq[w^{\prime}]\in V\mathcal{T}$ are such that $||w||_{A}=||w^{\prime}||_{A}=n\geq 0$. If there exists a Whitehead move $\tau\in\mathcal{W}_{N}$ such that $\tau([w])=[w^{\prime}]$ (and hence $\tau^{-1}[w^{\prime}]=[w]$, with $\tau^{-1}\in\mathcal{W}_{N}$) there is a topological edge $e$ connecting $[w]$ and $[w^{\prime}]$. There are two possible orientations on $e$ resulting in mutually inverse oriented edges: the edge with the orientation from $[w]$ to $[w^{\prime}]$ is labelled by $\tau$, and the edge $e$ with the orientation from $[w^{\prime}]$ to $[w]$ is labelled by $\tau^{-1}$.

Also, for $n\geq 0$ denote by $\mathcal{T}_{n}$ the subgraph of $\mathcal{T}$ spanned by all vertices $[w]\in V\mathcal{T}$ with $||w||_{A}=n$. For a vertex $[w]$ of $\mathcal{T}_{n}$ denote by $\mathcal{T}_{n}[w]$ the connected component of $\mathcal{T}_{n}$ containing $[w]$.

We first state the following simplified version of Whitehead’s “peak reduction” lemma (see [29, Proposition 1.2]):

Proposition 2.4**.**

The following hold:

1. (1)

An element $[w]\in\mathcal{C}_{N}$ is ${\rm Out}(F_{N})$-minimal if and only if $[w]$ is Whitehead-minimal. (Thus if $[w]$ is not ${\rm Out}(F_{N})$-minimal then there exists $\tau\in\mathcal{W}_{N}$ such that $||\tau(w)||_{A}<||w||_{A}$). 2. (2)

Suppose that $[w]\neq[w^{\prime}]$ are both ${\rm Out}(F_{N})$-minimal. Then ${\rm Out}(F_{N})[w]={\rm Out}(F_{N})[w^{\prime}]$ if and only if $||w||_{A}=||w^{\prime}||_{A}=n\geq 0$, and there exists a finite sequence $\tau_{1},\dots\tau_{k}\in\mathcal{W}_{N}$ such that $\tau_{k}\dots\tau_{1}[w]=[w^{\prime}]$ and that for $i=1,\dots,k$ we have

[TABLE]

Proposition 2.4 implies that if $[w]\in\mathcal{C}_{N}$ is ${\rm Out}(F_{N})$-minimal with $||w||_{A}=n$ then $\mathcal{M}([w])=\mathcal{T}_{n}[w]$.

We also record the following more general version of ”peak reduction”:

Proposition 2.5**.**

[37, Proposition 4.17]** Let $[w],[w^{\prime}]\in\mathcal{C}_{N}$ and $\varphi\in{\rm Out}_{N}$ be such that $[w^{\prime}]=\varphi([w])$ and that $||w^{\prime}||_{A}\leq||w||_{A}$. Then there exists a factorization $\varphi=\tau_{k}\dots\tau_{1}$ in ${\rm Out}(F_{N})$, where $\tau_{i}\in\mathcal{W}$ and where $||\tau_{i}\dots\tau_{1}w||_{A}\leq||w||_{A}$ for $i=1,\dots,k$.

Definition 2.6** (Whitehead algorithm).**

Let $F_{N}=F(A)$ be free of rank $N\geq 2$, with a fixed free basis $A$.

$\bullet$ The Whitehead minimization algorithm is the following process. Given $[w]\in\mathcal{C}_{N}$ put $[w_{1}]=[w]$. If $[w_{i}]$ is already constructed, check if there exists $\tau\in\mathcal{W}_{N}$ such that $||\tau(w_{i})||_{A}<||w_{i}||_{A}$. If not, declare that $[w_{i}]\in\mathcal{M}([w])$ (that is $[w_{i}]$ is an ${\rm Out}(F_{N})$-minimal element in ${\rm Out}(F_{N})[w]$ and terminate the algorithm. Put $[w_{i+1}]=[\tau(w_{i})]$.

$\bullet$ The Whitehead stabillization algorithm is the following process. Suppose that $[w]\in\mathcal{C}_{N}$ is Whitehead-minimal (and therefore ${\rm Out}(F_{N})$-minimal) with $||w||_{A}=n\geq 0$. Construct the component $\mathcal{T}_{n}([w])$ of $\mathcal{T}_{n}$ using the “breadth-first” stabilization process. Start with $S_{1}=\{[w]\}$. Now if a finite collection $S_{i}$ of conjugacy classes with $||.||_{A}=n$ is already constructed, for each element $[u]\in S_{i}$ and each $\tau\in\mathcal{W}_{N}$, put

[TABLE]

Terminate the process with the output $S_{i}$ for the smallest $i\geq 1$ such that $S_{i+1}=S_{i}$. Declare that $S_{i}=V\mathcal{T}_{n}([w])=\mathcal{M}([w])$.

$\bullet$ The Whitehead algorithm is the following process. Given $[w],[w^{\prime}]\in\mathcal{C}_{N}$, first apply the Whitehead minimization process to each of $[w],[w^{\prime}]$ to output elements $[u],[u^{\prime}]$ accordingly. Declare that $[u]\in\mathcal{M}([w])$ and $[u^{\prime}]\in\mathcal{M}([w^{\prime}])$. If $||u||_{A}\neq||u^{\prime}||_{A}$, declare that ${\rm Out}(F_{N})[w]\neq{\rm Out}(F_{N})[w^{\prime}]$ and terminate the process. Suppose that $||u||_{A}=||u^{\prime}||_{A}=n\geq 1$. Apply the Whitehead stabilization algorithm to $[u]$ to produce the set $S$. Declare that $S=\mathcal{T}_{n}([u])=\mathcal{M}([w])$. Then check wither $[u^{\prime}]\in S$. If $[u^{\prime}]\in S$, declare that ${\rm Out}(F_{N})[w]={\rm Out}(F_{N})[w^{\prime}]$, and if $[u^{\prime}]\not\in S$, declare that ${\rm Out}(F_{N})[w]\neq{\rm Out}(F_{N})[w^{\prime}]$, and terminate the process.

Remark 2.7*.*

Part (1) of Proposition 2.4 implies that the Whitehead minimization algorithm on an input $[w]\in\mathcal{C}_{N}$ always terminates in $O(||w||_{A}^{2})$ time (where we assume that $[w]$ is given to us as a cyclically reduced word in $F(A)$) and indeed outputs an element of $\mathcal{M}([w])$. The quadratic time bound arises since going from $[w_{i}]$ to $[w_{i+1}]$ takes a priori linear time in $||w_{i}||_{A}$, and since $||w_{1}||_{A}>||w_{2}||_{A}>\dots$, the process terminates with some $[w_{i}]$ with $i\leq||w||_{A}$.

Part (2) of Proposition 2.4 implies that the Whitehead minimization algorithm on an ${\rm Out}(F_{N})$-minimal input $[w]$ in $F_{N}=F(A)$ with $||w||_{A}=n$, always terminates in $O(\#V\mathcal{T}_{n}([w]))$ time, and indeed outputs the set $\mathcal{M}([w])=\mathcal{T}_{n}([w])$. Taken together, Proposition 2.4 implies that the Whitehead algorithm on the input $[w],[w^{\prime}]\in\mathcal{C}_{N}$ does correctly decide whether or not ${\rm Out}(F_{N})[w]={\rm Out}(F_{N})[w^{\prime}]$.

Overall, the a priori worst-case complexity of Whitehead’s algorithm on the input $[w],[w^{\prime}]$ is exponential in $\max\{||w||_{A},||w^{\prime}||_{A}\}$ because for $[u]\in V\mathcal{T}_{n}$ the cardinality $\#V\mathcal{T}_{n}([u])$ is at most exponential in $n$.

Definition 2.8**.**

Suppose that $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ is a fixed finite set of “auxiliary” automorphisms.

• •

The $\mathfrak{W}$-speed-up of the Whitehead minimization algorithm consists in taking the input $[w]\neq 1$, computing $\mathfrak{W}[w]=\{\psi[w]|\psi\in\mathfrak{W}\}$ first and then applying the Whitehead minimization algorithm, in parallel to $[w]$ and each of the elements of $\mathfrak{W}([w])$. The result is again an element of $\mathcal{M}([w])$.

• •

The $\mathfrak{W}$-speed-up of the Whitehead’s algorithm consists in doing the following. Given $[w],[w^{\prime}]\neq 1$, first apply the $\mathfrak{W}$-speed-up of the Whitehead minimization algorithm to both $[w]$ and $[w^{\prime}]$ to find $[u]\in\mathcal{M}([w])$ and $[u^{\prime}]\in\mathcal{M}([w^{\prime}])$. Then proceed exactly as in Whitehead’s algorithm to decide whether or not ${\rm Out}(F_{N})[w]={\rm Out}(F_{N})[w^{\prime}]$.

Since $\mathfrak{W}$ is finite an fixed, the a priori complexity estimates for these speed-up versions are the same as in Remark 2.7, although with worse multiplicative constants.

3. $(M,\lambda,\varepsilon)$-minimality and Whitehead’s algorithm

Let $F_{N}=F(A)$ be free of rank $N\geq 2$ where $A=\{a_{1},\dots,a_{N}\}$ is a fixed free basis of $F_{N}$.

3.1. Main definitions

Definition 3.1**.**

Let $M\geq 1$ be an integer, let $\lambda>1$ and let $0\leq\varepsilon<\lambda-1$.

A finite set $S$ of conjugacy classes of nontrivial elements of $F_{N}$ is called $(M,\lambda,\varepsilon)$-minimizing if it satisfies the following properties:

1. (1)

We have $\#(S)\leq M$. 2. (2)

For any $[u],[u^{\prime}]\in S$ we have ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$. 3. (3)

For any $[u],[u^{\prime}]\in S$ we have $1-\varepsilon\leq\frac{||u^{\prime}||_{A}}{||u||_{A}}\leq 1+\varepsilon$. 4. (4)

For every $[u]\in S$ and every $\varphi\in{\rm Out}(F_{N})$ such that $\varphi([u])\not\in S$ we have $\frac{||\varphi(u)||_{A}}{||u||_{A}}\geq\lambda>1+\varepsilon$.

In this case for any $[u]\in S$ we also say that $S$ is a $(M,\lambda,\varepsilon)$-minimizing set for $[u]$.

We say that a nontrivial conjugacy class $[u]$ in $F_{N}$ is $(M,\lambda,\varepsilon)$-minimal if there exists an $(M,\lambda,\varepsilon)$-minimizing set $S$ for $[u]$ (and thus $[u]\in S$).

Note that if $S$ is a $(M,\lambda,\varepsilon)$-minimizing set and if $[u]\in S$ then for $\varphi\in{\rm Out}(F_{N})$ either $\varphi(u)\in S$ or $\frac{||\varphi(u)||_{A}}{||u||_{A}}\geq\lambda$, and these outcomes are mutually exclusive.

We record the following useful immediate corollary of the above definition:

Lemma 3.2**.**

Let $M\geq 1$ be an integer, let $\lambda>1$, let $0\leq\varepsilon<\lambda-1$ and let $S$ be an $(M,\lambda,\varepsilon)$-minimizing set of conjugacy classes in $F_{N}$. Then for any $[u]\in S$ and $\varphi\in{\rm Out}(F_{N})$ such that $||\varphi(u)||_{A}\leq(1+\varepsilon)||u||_{A}$ we have $\varphi([u])\in S$.

∎

Definition 3.3**.**

Let $M\geq 1$ be an integer, let $\lambda>1$ and let $0\leq\varepsilon<\lambda-1$.

A finite set $S\subseteq\mathcal{C}_{N}$ of conjugacy classes of nontrivial elements of $F_{N}$ is called $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing if it satisfies the following properties:

1. (1)

We have $\#(S)\leq M$. 2. (2)

For any $[u],[u^{\prime}]\in S$ we have ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$. 3. (3)

For any $[u],[u^{\prime}]\in S$ we have $1-\varepsilon\leq\frac{||u^{\prime}||_{A}}{||u||_{A}}\leq 1+\varepsilon$. 4. (4)

For any $[u]\in S$ and $\tau\in\mathcal{W}_{N}$ exactly one of the following occurs:

• (i)

We have $\tau([u])\in S$.

• (ii)

We have $\tau([u])\not\in S$ and $\frac{||\tau(u)||_{A}}{||u||_{A}}\geq\lambda>1+\varepsilon$.

In this case for any $[u]\in S$ we also say that $S$ is a $(M,\lambda,\varepsilon,\mathcal{W})$-minimizing set for $[u]$.

We say that a nontrivial conjugacy class $[u]$ in $F_{N}$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal if there exists an $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing set $S$ for $[u]$ (and thus $[u]\in S$).

Lemma 3.4**.**

Let $M\geq 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be such that $\varepsilon<\lambda-1$ and $\lambda(1-\varepsilon)>1$. Let $S\subseteq\mathcal{C}_{N}$ be a finite set of conjugacy classes of nontrivial elements of $F_{N}$ such that $S$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing.

1. (1)

For any $[u]\in S$ and $\tau\in\mathcal{W}_{N}$ such that $||\tau(u)||_{A}\leq(1+\varepsilon)||u||_{A}$ we have $\tau([u])\in S$. 2. (2)

For any $[u]\in S$ and $\varphi\in{\rm Out}(F_{N})$ such that $||\varphi(u)||_{A}\leq||u||_{A}$ we have $\varphi([u])\in S$.

Proof.

Part (1) follows from conditions (3), (4) of Definition 3.3.

For (2), suppose that $[u]\in S$ and $\varphi\in{\rm Out}(F_{N})$ are such that $||\varphi(u)||_{A}\leq||u||_{A}$. By Proposition 2.5, there exist $\tau_{1},\dots,\tau_{k}\in\mathcal{W}_{N}$ such that $\varphi=\tau_{k}\dots\tau_{1}$ and that for $[u_{0}]=[u]$, $[u_{i}]=\tau_{i}\dots\tau_{1}([u])$ for $i=1,\dots,k$ we have $||u_{i}||_{A}\leq||u||_{A}$. Note that $[u_{k}]=\varphi([u])$.

We argue by induction on $i$ that $[u_{i}]\in S$ for $i=1,\dots,k$. We have $[u]=[u_{0}]\in S$. Suppose now $0\leq i<k$ and $[u_{i}]\in S$. We need to show that $[u_{i+1}]=\tau_{i+1}[u_{i}]\in S$. Suppose, on the contrary, that $[u_{i+1}]\not\in S$. Then $||u_{i+1}||_{A}/||u_{i}||_{A}\geq\lambda$. Since $[u],[u_{i}]\in S$, also have $||u_{i}||_{A}/||u||_{A}\geq 1-\varepsilon$. Therefore $||u_{i+1}||_{A}/||u||_{A}\geq\lambda(1-\varepsilon)$, so that $||u_{i+1}||_{A}\geq\lambda(1-\varepsilon)||u||_{A}>||u||_{A}$ since $\lambda(1-\varepsilon)>1$. This contradicts the choice of $\tau_{1},\dots,\tau_{k}$. Thus $[u_{i+1}]\in S$, as required.

Hence $[u_{k}]=\varphi([u])\in S$, and part (2) of the lemma is verified.

∎

The definitions directly imply that an $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing set $S$ is $(M,\lambda,\varepsilon)$-minimizing. It turns out that the converse also holds, but with slightly smaller $\varepsilon$ and slightly bigger $\lambda$.

Proposition 3.5**.**

Let $M\geq 1$ be an integer, let $\lambda>1$ and let $0\leq\varepsilon<\lambda-1$. Let $0<\varepsilon^{\prime}<\varepsilon$ and $\lambda^{\prime}>\lambda>1$ be such that be such that $\lambda^{\prime}(1-\varepsilon^{\prime})>\lambda$.

Let $S\subseteq\mathcal{C}_{N}$ be a finite set of conjugacy classes of nontrivial elements of $F_{N}$ be such that $S$ is $(M,\lambda^{\prime},\varepsilon^{\prime},\mathcal{W}_{N})$-minimizing.

Then $S$ is $(M,\lambda,\varepsilon)$-minimizing.

Proof.

We need to verify that conditions (1)-(4) of Definition 3.1 of an $(M,\lambda,\varepsilon)$-minimizing set hold for $S$.

Since $S$ is $(M,\lambda^{\prime},\varepsilon^{\prime},\mathcal{W}_{N})$-minimizing, it follows that $\#(S)\leq M$, any two elements of $S$ are in the same ${\rm Out}(F_{N})$-orbit, and for any $[u],[u^{\prime}]\in S$ we have $\frac{||u^{\prime}||_{A}}{||u||_{A}}\in[1-\varepsilon^{\prime},1+\varepsilon^{\prime}]\subseteq[1-\varepsilon,1+\varepsilon]$. Thus we only need to verify condition (4) of Definition 3.1 for $S$.

Let $[u]\in S$ and let $\psi\in{\rm Out}(F_{N})$ be such that $\varphi([u])\not\in S$. Part (2) of Lemma 3.4 implies that $||u||_{A}<||\varphi(u)||_{A}$. Therefore by Proposition 2.5, there exist $\tau_{1},\dots,\tau_{k}\in\mathcal{W}_{N}$ such that $\varphi=\tau_{k}\dots\tau_{1}$ and that for $[v_{0}]=[u]$, $[v_{i}]=\tau_{i}\dots\tau_{1}([u])$ for $i=1,\dots,k$ we have $||v_{i}||_{A}\leq||\varphi(u)||_{A}$. Note that $[v_{k}]=\varphi([u])$. Since $[v_{k}]\not\in S$, the set $\{i\geq 0|[v_{i}]\not\in S\}$ is nonempty. Put $j=\min\{i\geq 0|[v_{i}]\not\in S\}$. Since $[v_{0}]=[u]\in S$, we have $j\geq 1$, and $[v_{i}]\in S$ for all $0\leq i<j$. Since $[v_{j-1}],[u]\in S$, we have $||v_{j-1}||_{A}\geq(1-\varepsilon^{\prime})||u||_{A}$. Since $[v_{j-1}]\in S$ and $[v_{j}]=\tau_{j}([v_{j-1}])\not\in S$, it follows that $||v_{j}||_{A}\geq\lambda||v_{j-1}||_{A}$ and therefore $||v_{j}||_{A}\geq\lambda(1-\varepsilon^{\prime})||u||_{A}$. We also have $||\varphi(u)||_{A}\geq||v_{j}||_{A}$ and hence

[TABLE]

Thus the set $S$ is $(M,\lambda,\varepsilon)$-minimizing, as required. ∎

Definition 3.6**.**

Let $M\geq 1$ be an integer, let $\lambda>1$ and let $0\leq\varepsilon<\lambda-1$.

Let $1\neq w\in F_{N}$. We say that $[w]$ is $(M,\lambda,\varepsilon)$-minimizable in $F_{N}=F(A)$ if there exists a subset $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ such that $\#(\mathfrak{W})\leq M$ and that the set $S=\mathfrak{W}[w]$ is $(M,\lambda,\varepsilon)$-minimizing (or, equivalently, if the orbit ${\rm Out}(F_{N})[w]$ contains a $(M,\lambda,\varepsilon)$-minimal element). In this case we say that $\mathfrak{W}$ is $(M,\lambda,\varepsilon)$-reducing for $[w]$.

Note that for $1\neq w\in F_{N}$ the conjugacy class $[w]$ is $(M,\lambda,\varepsilon)$-minimizable if and only if the orbit ${\rm Out}(F_{N})[w]$ contains a $(M,\lambda,\varepsilon)$-minimal element.

3.2. Behavior of Whitehead’s algorithm

We now have:

Proposition 3.7**.**

Let $\lambda>1$, let $0\leq\varepsilon<\lambda-1$, let $[u]\in\mathcal{C}_{N}$ be a $(M,\lambda,\varepsilon)$-minimal element and let $S\subseteq\mathcal{C}_{N}$ be an $(M,\lambda,\varepsilon)$-minimizing set for $[u]$ (so that $[u]\in S$). Then the following hold:

1. (1)

We have $\mathcal{M}([u])\subseteq S$, and, in particular, $\#\mathcal{M}([u])\leq M$. 2. (2)

For every $[u^{\prime}]\in\mathcal{M}([u])$ we have $1\leq\frac{||u||_{A}}{||u^{\prime}||_{A}}\leq 1+\varepsilon$. 3. (3)

For every $[u^{\prime}]\in\mathcal{M}([u])$ we have $\mathcal{M}([u])=V\mathcal{T}_{n}([u^{\prime}])$ where $n=||u^{\prime}||_{A}$. 4. (4)

If $\tau\in\mathcal{W}_{N}$ is a Whitehead automorphism such that $||\tau(u)||_{A}<||u||_{A}$ then $\tau([u])\in S$. 5. (5)

If $\tau_{1},\tau_{2},\dots\tau_{k}\in\mathcal{W}_{N}$ are such that

[TABLE]

then $k\leq M-1$ and we have $[u_{i}]:=\tau_{i}\dots\tau_{1}([u])\in S$ for $i=1,\dots,k$. 6. (6)

If such a sequence $\tau_{1},\tau_{2},\dots\tau_{k}\in\mathcal{W}_{N}$ as in (4) is such that is $[u_{k}]$ is $\mathcal{W}_{N}$-minimal then $[u_{k}]\in\mathcal{M}([u])$.

Proof.

Parts (1) and (2) follow directly from Definition 3.1. Part (3) holds by the general peak reduction properties of Whitehead’s algorithm. Part (4) follows from property (4) in Definition  3.1. Suppose now that $\tau_{1},\tau_{2},\dots\tau_{k}\in\mathcal{W}_{N}$ are as in part (5) of the proposition. Since the cyclically reduced lengths $||u||_{A}>||u_{1}||_{A}>\dots>||u_{k}||_{A}$ are strictly decreasing, the conjugacy classes $[u],[u_{1}],\dots[u_{k}]$ are distinct. Since by assumption $[u]\in S$, part (4) of the proposition implies that $[u],[u_{1}],\dots,[u_{k}]\in S$. Since $\#S\leq M$, it follows that $k\leq M-1$.

Part (6) follows from part (5) since every $\mathcal{W}_{N}$-minimal conjugacy class is ${\rm Out}(F_{N})$-minimal. ∎

Proposition 3.7 then directly implies:

Corollary 3.8**.**

Let $\lambda>1$, let $0\leq\varepsilon<\lambda-1$ and let $[w]\in\mathcal{C}_{N}$ be $(M,\lambda,\varepsilon)$-minimizable, with a $(M,\lambda,\varepsilon)$-reducing for $[w]$ set $\mathfrak{W}\subseteq{\rm Out}(F_{N})$. Let $S=\mathfrak{W}[w]$ (so that $S$ is $(M,\lambda,\varepsilon)$-minimizing). Then:

1. (1)

We have $\mathcal{M}([w])\subseteq S$, and, in particular, $\#\mathcal{M}([w])\leq M$. 2. (2)

For every $[u^{\prime}]\in\mathcal{M}([w])$ we have $\mathcal{M}([w])=V\mathcal{T}_{n}([u^{\prime}])$ where $n=||u^{\prime}||_{A}$.

∎

Definition 3.9**.**

Let $M\geq 1$, $\lambda>1$, and $0\leq\varepsilon<\lambda-1$.

1. (1)

We denote by $U_{N}(M,\lambda,\varepsilon)$ the set of all $1\neq u\in F_{N}$ such that $[u]$ is $(M,\lambda,\varepsilon)$-minimal. 2. (2)

We denote by $Y_{N}(M,\lambda,\varepsilon)$ the set of all $1\neq w\in F_{N}$ such that there exists $[u]\in{\rm Out}(F_{N})[w]$ such that $[u]$ is $(M,\lambda,\varepsilon)$-minimal. [That is, $Y_{N}(M,\lambda,\varepsilon)$ is the set of all $1\neq w\in F_{N}$ such that $[w]$ is $(M,\lambda,\varepsilon)$-minimizable.] 3. (3)

Let $\psi\in{\rm Out}(F_{N})$. Denote by $U_{N}(M,\lambda,\varepsilon;\psi)$ the set of all $1\neq w\in F_{N}$ such that $\psi([w])$ is $(M,\lambda,\varepsilon)$-minimal.

Lemma 3.10**.**

Let $u\in Y_{N}(M,\lambda,\varepsilon)$ and let $[u^{\prime}]\in\mathcal{M}([u])$ (that is, $[u^{\prime}]$ is an ${\rm Out}(F_{N})$-minimal element in the orbit ${\rm Out}(F_{N})[u]$). Then $u^{\prime}\in U_{N}(M,\lambda,\varepsilon)$ (that is, $[u^{\prime}]$ is $(M,\lambda,\varepsilon)$-minimal).

Proof.

Since $u\in Y_{N}(M,\lambda,\varepsilon)$, there exists $\varphi\in{\rm Out}(F_{N})$ such that $\varphi([u])$ is $(M,\lambda,\varepsilon)$-minimal, so that $\varphi([u])$ belongs to some $(M,\lambda,\varepsilon)$-minimizing set $S$. Part (1) of Proposition 3.7 implies that $\mathcal{M}(\varphi([u]))\subseteq S$, so that every element of $\mathcal{M}(\varphi([u]))$ is $(M,\lambda,\varepsilon)$-minimal. Since $\mathcal{M}(\varphi([u]))=\mathcal{M}([u])$, the statement of the lemma follows. ∎

We now summarize algorithmic properties of $(M,\lambda,\varepsilon)$-minimal in relation to Whitehead’s algorithm.

Theorem 3.11**.**

Let $M\geq 1$, $\lambda>1$, and $0\leq\varepsilon<\lambda-1$. Then there exists a constant $K\geq 1$ such that the following hold:

1. (a)

For any $u\in U_{N}(M,\lambda,\varepsilon)$ the Whitehead minimization algorithm on the input $u$ terminates time $\leq K|u|_{A}$ and produces an element of $\mathcal{M}([u])$. 2. (b)

For any $u_{1},u_{2}\in U_{N}(M,\lambda,\varepsilon)$, the Whitehead algorithm for the automorphic equivalence problem in $F_{N}$ terminates in time at most $K\max\{|u_{1}|_{A},|u_{2}|_{A}\}$, on the input $(u_{1},u_{2})$. 3. (c)

For any $u_{1}\in Y_{N}(M,\lambda,\varepsilon)$ and any $1\neq u_{2}\in F_{N}$, the Whitehead algorithm for the automorphic equivalence problem in $F_{N}$ terminates in time at most $K\max\{|u_{1}|_{A}^{2},|u_{2}|_{A}^{2}\}$, on the input $(u_{1},u_{2})$. 4. (d)

For any $u_{1}\in U_{N}(M,\lambda,\varepsilon)$ and any $1\neq u_{2}\in F_{N}$, the Whitehead algorithm for the automorphic equivalence problem in $F_{N}$ terminates in time $K\max\{|u_{1}|_{A},|u_{2}|_{A}^{2}\}$, on the input $(u_{1},u_{2})$. 5. (e)

Let $\psi\in{\rm Out}(F_{N})$ be a fixed element. Then there is $K^{\prime}=K^{\prime}\geq 1$ such that for any $u_{1},u_{2}\in U_{N}(M,\lambda,\varepsilon;\psi)$, the $\psi$-speed-up of Whitehead’s algorithm decides in time at most $K^{\prime}\max\{|u_{1}|_{A},|u_{2}|_{A}\}$, whether or not ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$. 6. (f)

Let $\psi\in{\rm Out}(F_{N})$ be a fixed element. Then there is $K^{\prime}=K^{\prime}\geq 1$ such that for any $u_{1}\in U_{N}(M,\lambda,\varepsilon;\psi)$ and any $1\neq u_{2}\in F_{N}$, the $\psi$-speed-up of Whitehead’s algorithm decides in time at most $K^{\prime}\max\{|u_{1}|_{A},|u_{2}|_{A}^{2}\}$, whether or not ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$.

Proof.

(a) Let $u\in U_{N}(M,\lambda,\varepsilon)$ be arbitrary. Let $S$ be an $(M,\lambda,\varepsilon)$-minimizing set containing $[u]$. Thus $\#S\leq M$. By Lemma 3.2, if $[v]\in S$ and $\tau\in\mathcal{W}_{N}$ is a Whitehead move such that $||\tau(v)||_{A}<||v||_{A}$ then $\tau([v])\in S$. Therefore starting with $u$ and iteratively looking for Whitehead moves that decrease the $||.||_{A}$-length terminates after a chain of $\leq M$ such moves with a conjugacy class that is Whitehead-minimal and therefore is ${\rm Out}(F_{N})$-minimal, that is, an element of $\mathcal{M}([u])$. This process takes at most time $C_{1}|u|_{A}$ for some constant $C_{1}>0$ depending only on $N,M,\lambda,\varepsilon$.

(b) Let $u_{1},u_{2}\in U_{N}(M,\lambda,\varepsilon)$ so that $[u_{1}],[u_{2}]$ are $(M,\lambda,\varepsilon)$-minimal. By part (a) above, applying the Whitehead minimization algorithm to $[u_{i}]$ terminates in at most $M$ steps with an ${\rm Out}(F_{N})$-minimal element $[u_{i}^{\prime}]$ such that $n_{i}=||u_{i}^{\prime}||_{A}\leq||u_{i}||_{A}\leq|u_{i}|_{A}$. Each of these $\leq M$ takes at most linear times in $|u_{i}|_{A}$ since the number of whitehead automorphisms in $\mathcal{W}_{N}$ is finite and fixed. Thus it takes linear time in $|u_{i}|_{A}$ to produce $[u_{i}^{\prime}]\in\mathcal{M}([u_{i}])$. If $n_{1}\neq n_{2}$ then ${\rm Out}(F_{N})[u_{1}]\neq{\rm Out}(F_{N})[u_{2}]$ and we are done. Suppose that $n=n_{1}=n_{2}$. By part (3) of Proposition 3.7 we have $\mathcal{M}([u_{i}])=V\mathcal{T}_{n}([u_{i}^{\prime}])$ for $i=1,2$. Moreover, by part (1) of Proposition 3.7 we have $\#V\mathcal{T}_{n}([u_{i}^{\prime}])\leq M$ here. Since $M$ is fixed, it takes linear time in $|u_{i}|_{A}$ to construct the graph $\mathcal{T}_{n}([u_{i}^{\prime}])\leq M$ from $u_{i}^{\prime}$. Then ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$ if and only if $\#V\mathcal{T}_{n}([u_{1}^{\prime}])\cap\#V\mathcal{T}_{n}([u_{2}^{\prime}])\neq\varnothing$, and this condition can be checked in linear time in $\max\{|u_{1}|_{A},|u_{2}|_{A}\}$. Summing up we get that the total running time of the Whitehead algorithm for the automorphic equivalence problem in $F_{N}$ is time at most $C_{2}\max\{|u_{1}|_{A},|u_{2}|_{A}\}$, for some constant $C_{2}>0$ depending only on $N,M,\lambda,\varepsilon$.

(c) Now suppose that $u_{1}\in Y_{N}(M,\lambda,\varepsilon)$ and $1\neq u_{2}\in F_{N}$. We first apply the Whitehead minimization algorithm to each of $u_{1},u_{2}$ to find $Out(F_{N})$-minimal elements $[u_{i}^{\prime}]\in Out(F_{N})[u_{i}]$ for $i=1,2$. Producing $u_{i}^{\prime}$ from $u_{i}$ takes quadratic time in terms of $|u_{i}|_{A}$. Note that by Lemma 3.10 the element $[u_{1}^{\prime}]$ is $(M,\lambda,\varepsilon)$-minimal, that is $u_{1}^{\prime}\in U_{N}(M,\lambda,\varepsilon)$.

Again put $n_{i}=||u_{i}|_{A}$. If $n_{1}\neq n_{2}$ then ${\rm Out}(F_{N})[u_{1}]\neq{\rm Out}(F_{N})[u_{2}]$ and we are done. Suppose that $n=n_{1}=n_{2}$. Since $u_{1}^{\prime}$ is ${\rm Out}(F_{N})$-minimal and $(M,\lambda,\varepsilon)$-minimal, by parts (3) and (1) of Proposition 3.7 we have $\mathcal{M}([u_{1}])=\mathcal{M}([u_{1}^{\prime}])=V\mathcal{T}_{n}([u_{1}^{\prime}])$ and $\#V\mathcal{T}_{n}([u_{1}^{\prime}])\leq M$. Then, since $M$ is fixed, it takes at most linear time in $n=||u_{1}^{\prime}||_{A}\leq|u_{1}|_{A}$ to construct the graph $\mathcal{T}_{n}([u_{1}^{\prime}])$. Recall also that $[u_{2}^{\prime}]\in\mathcal{M}([u_{2}])$ and $||u_{2}||_{A}=n$. Then we have ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$ if and only if $[u_{2}^{\prime}]\in\mathcal{T}_{n}([u_{1}^{\prime}])$. This last condition can be checked in linear time in $n$. Again, summing up we see that the total running time of the Whitehead algorithm on $(u_{1},u_{2})$ is at most $C_{3}\max\{|u_{1}|_{A}^{2},|u_{2}|_{A}^{2}\}$, for some constant $C_{3}>0$ depending only on $N,M,\lambda,\varepsilon$.

(d) Now let $u_{1}\in U_{N}(M,\lambda,\varepsilon)$ and $1\neq u_{2}\in F_{N}$. We first apply the Whitehead minimization algorithm to each of $u_{1},u_{2}$ to find $Out(F_{N})$-minimal elements $[u_{i}^{\prime}]\in Out(F_{N})[u_{i}]$ for $i=1,2$. As in (b), producing $u_{1}^{\prime}$ from $u_{1}$ takes linear time in $|u_{1}|_{A}$, because $u_{1}$ is $(M,\lambda,\varepsilon)$-minimal. Producing $u_{2}^{\prime}$ from $u_{2}$ takes at most quadratic time in $|u_{2}|_{A}$, by the general Whitehead’s minimization algorithm properties. After that we proceed exactly in (2) above to decide if $[u_{1}^{\prime}]$ and $[u_{2}^{\prime}]$ are ${\rm Out}(F_{N})$-equivalent. Summing up we see that the total running time of the Whitehead algorithm on $(u_{1},u_{2})$ is at most $C_{4}\max\{|u_{1}|_{A},|u_{2}|_{A}^{2}\}$ in this case, for some constant $C_{4}>0$ depending only on $N,M,\lambda,\varepsilon$.

(e) Choose an automorphism $\Psi\in{\rm Aut}(F_{N})$ in the outer automorphism class $\psi$ and put $C=\max_{i=1}^{N}|\Psi(a_{i})|_{A}$. Since $\varphi$ and $\Psi$ are fixed, given $u_{1},u_{2}\in U_{N}(M,\lambda,\varepsilon;\psi)$, for $i=1,2$ it takes linear time in $|u_{i}|_{A}$ to compute the element $\Psi(u_{i})$, and $|\Psi(u_{i})|_{A}\leq C|u_{i}|_{A}$. Moreover, the assumption that $u_{1},u_{2}\in U_{N}(M,\lambda,\varepsilon;\psi)$ implies that $\Psi(u_{1}),\Psi(u_{2})$ are $(M,\lambda,\varepsilon)$-minimal. Then by part (b) above, the Whitehead algorithm on the input $(\Psi(u_{1}),\Psi(u_{2}))$ terminates in linear time in $C\max\{|u_{1}|_{A},|u_{2}|_{A}\}$ and decides whether or not ${\rm Aut}(F_{N})\Psi(u_{1})={\rm Aut}(F_{N})\Psi(u_{2})$, that is, whether or not ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$. The total running time required for this process is at most $C_{5}\max\{|u_{1}|_{A},|u_{2}|_{A}\}$, for some constant $C_{5}>0$ depending only on $N,M,\lambda,\varepsilon$ and $\Psi$.

(f) We chose $\Psi\in{\rm Aut}(F_{N})$ and $C>0$ as in the proof of part (e) above. Given any $u_{1}\in U_{N}(M,\lambda,\varepsilon;\psi)$ and any $1\neq u_{2}\in F_{N}$ we first compute, in linear time in $|u_{1}|_{A}$, the element $\Psi(u_{1})$ and again observe that $\Psi(u_{1})$ is $(M,\lambda,\varepsilon)$-minimal and that $|\Psi(u_{1})|_{A}\leq C|u_{1}|_{A}$. We then apply to the pair $(\Psi(u_{1}),u_{2})$ the algorithm from part (d) of this proposition to decide whether or not ${\rm Out}(F_{N})[u_{1}]={\rm Out}(F_{N})[u_{2}]$. The overall running time is of this process is at most $C_{6}\max\{|u_{1}|_{A},|u_{2}|_{A}^{2}\}$, for some constant $C_{6}>0$ depending only on $N,M,\lambda,\varepsilon$ and $\Psi$.

∎

We recall another basic fact related to Whitehead’s algorithm which describes $Out(F_{N})$-stabilizers of conjugacy classes in $F_{N}$:

Proposition 3.12**.**

Let $1\neq u\in F_{N}$ be such that $[u]$ is $Out(F_{N})$-minimal, and let $n=||u||_{A}$. Then for $\varphi\in{\rm Out}(F_{N})$ we have $\varphi([u])=[u]$ if and only if there exists a sequence of Whitehead automorphisms $\tau_{1},\dots,\tau_{k}\in\mathcal{W}_{N}$ such that for $u_{i}=\tau_{i}\dots\tau_{1}([u])$, we have $||u_{i}||_{A}=n$ for $i=1,\dots,k$ and $[u_{k}]=[u]$ and such that $\varphi=\tau_{k}\dots\tau_{1}$ in ${\rm Out}(F_{N})$.

Recall that for $n\geq 1$ the oriented edges of the graph $\mathcal{T}_{n}$ are labelled by Whitehead moves $\tau\in\mathcal{W}_{N}$. Thus oriented edge-paths in $\mathcal{T}_{n}$ are labelled by products of Whitehead moves. Recall also that for a vertex $[u]$ of $\mathcal{T}_{n}$, the graph $\mathcal{T}_{n}[u]$ is the connected component of $\mathcal{T}_{n}$ containing $[u]$. Therefore we get a natural labelling homomorphism $\rho_{[u]}:\pi_{1}(\mathcal{T}_{n}[u],[u])\to{\rm Out}(F_{N})$ where a closed loop at $[u]$ in $\mathcal{T}_{n}$ is mapped to the element of ${\rm Out}(F_{N})$ given by the label of this loop in $\mathcal{T}_{n}$. Note also that, since the set of possible edge labels $\mathcal{W}_{N}$ is finite, the rank of the free group $\pi_{1}(\mathcal{T}_{n},[u])$ is bounded above by some constant $K=K(N,\#V\mathcal{T}_{n}[u])$.

Proposition 3.12 now directly implies:

Corollary 3.13**.**

Let $1\neq u\in F_{N}$ be such that $[u]$ is $Out(F_{N})$-minimal, and let $n=||u||_{A}$. Then:

1. (1)

We have $Stab_{{\rm Out}(F_{N})}([u])=\rho_{[u]}\left(\pi_{1}(\mathcal{T}_{n}[u],[u])\right)$. 2. (2)

We have ${\rm rank}\ Stab_{{\rm Out}(F_{N})}([u])\leq K(N,\#V\mathcal{T}_{n}[u])$.

Proposition 3.14**.**

Let $M\geq 1$, $\lambda>1$, and $0\leq\varepsilon<\lambda-1$. Let $u\in Y_{N}(M,\lambda,\varepsilon)$ (that is $1\neq u\in F_{N}$ and the orbit ${\rm Out}(F_{N})[u]$ contains an $(M,\lambda,\varepsilon)$-minimal element). Then

[TABLE]

Proof.

Let $[u^{\prime}]\in{\rm Out}(F_{N})[u]$ be an $(M,\lambda,\varepsilon)$-minimal element. Let $[u^{\prime\prime}]$ be an ${\rm Out}(F_{N})$-minimal element in ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$ and put $n=||u^{\prime\prime}||_{A}$. Part (1) of Proposition 3.7 implies that $[u^{\prime\prime}]$ is also $(M,\lambda,\varepsilon)$-minimal and that $V\mathcal{T}_{n}[u^{\prime\prime}]=\mathcal{M}[u]=\mathcal{M}[u^{\prime}]=\mathcal{M}[u^{\prime\prime}]$ has cardinality $\leq M$. Then by Corollary 3.13 we have ${\rm rank}\ Stab_{{\rm Out}(F_{N})}([u])\leq K(N,M)$, as claimed. ∎

3.3. Algorithmic detectability

For a finite nonempty subset $S\subseteq\mathcal{C}_{N}$ denote $||S||_{A}=\max\{||u||_{A}|[u]\in S\}$.

Remark 3.15*.*

Let an integer $M\geq 1$ and rational numbers $0<\varepsilon<1$ and $\lambda>1+\varepsilon$ be fixed.

(1) Since the set $\mathcal{W}_{N}\subseteq{\rm Out}(F_{N})$ of Whitehead moves is finite and fixed, given a subset $S\subseteq\mathcal{C}_{N}-\{1\}$ of cardinality $\leq M$ we can check in linear time in $||S||_{A}$ whether or not $S$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing.

(2) In this setting, given such $S$ it is also possible to algorithmically check whether or not $(M,\lambda,\varepsilon)$-minimizing, but only with the exponential time, in terms of $||S||_{A}$, complexity estimate. Indeed, we first check (in linear time), if conditions (1) and (3) of Definition 3.1 of an $(M,\lambda,\varepsilon)$-minimizing set hold for $S$. We then use Whitehead’s algorithm to check if condition (2) of Definition 3.1 also holds. Suppose they do (otherwise $S$ is not $(M,\lambda,\varepsilon)$-minimizing). Then for every $[u]\in S$ compute, using Whitehead’s algorithm, the (finite) set $\mathcal{F}[u]=\{[u^{\prime}]\in{\rm Out}(F_{N})[u]|||u^{\prime}||_{A}\leq||u||_{A}\}$. For each $[u]\in S$ check whether $\mathcal{F}[u]\subseteq S$. If not then $S$ is not $(M,\lambda,\varepsilon)$-minimizing. So suppose that for all $[u]\in S$ $\mathcal{F}[u]\subseteq S$. Since balls in the Cayley graphs of $F_{N}=F(A)$ are finite and since $||w||_{A}\in\mathbb{Z}_{\geq 0}$ for all $w\in F_{N}$, we can then use Whitehead’s algorithm to compute, for each $[u]\in S$ the number \rho([u]):=\min\{\frac{||u^{\prime}||_{A}}{||u||_{A}}\big{|}[u^{\prime}]\in{\rm Out}(F_{N})[u]\text{ and }||u^{\prime}||_{A}>||u||_{A}\}. Then compute $\rho=\min_{[u]\in S}\rho([u])$. Then $S$ is $(M,\lambda,\varepsilon)$-minimizing if and only if $\rho\geq\lambda$. The complexity of this procedure for deciding if a subset $S\subseteq\mathcal{C}_{N}$ with $\#S\leq M$ is $(M,\lambda,\varepsilon)$-minimizing is exponential time in $||S||_{A}$ (when $M,\lambda,\varepsilon$ are fixed).

(3) We can then also decide, given $[u]\in\mathcal{C}_{N}$, whether or not $[u]$ is $(M,\lambda,\varepsilon)$-minimal, that is, whether or not $[u]$ belongs to some $(M,\lambda,\varepsilon)$-minimizing subset. Namely, list all subsets $S\subseteq\mathcal{C}_{N}$ of cardinality $\leq M$ containing $[u]$ and with $||S||_{A}\leq(1+\varepsilon)||u||_{A}$ and for each of them run the algorithm from (2) to decide if $S$ is $(M,\lambda,\varepsilon)$-minimizing. Again, since $M,\lambda,\varepsilon$ are fixed, this check can be done in exponential time in $||S||_{A}$.

It turns out that deciding whether an element $[u]$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal can be done in linear time in $||u||_{A}$ (under slightly more stringent assumptions in $\lambda,\varepsilon$).

Lemma 3.16**.**

Let $M\geq 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be such that $\varepsilon<\lambda-1$ and $\lambda\frac{1-\varepsilon}{1+\varepsilon}>1$. Let $S\subseteq\mathcal{C}_{N}$ be a finite set of conjugacy classes of nontrivial elements of $F_{N}$ such that $S$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing. Let $[u]\in S$.

Then for $[u^{\prime}]\in\mathcal{C}_{N},[u^{\prime}]\neq[u]$ the following conditions are equivalent:

1. (1)

We have $[u^{\prime}]\in S$. 2. (2)

There exists a chain $\tau_{1},\dots,\tau_{k}\in\mathcal{W}_{N}$ such that $k\leq M$, that $\tau_{k}\dots\tau_{1}[u]=[u^{\prime}]$ and that with $[u_{0}]=[u]$ $[u_{i}]=\tau_{i}\dots\tau_{1}[u]$ we have $||u_{i+1}||_{A}\leq(1+\varepsilon)||u_{i}||_{A}$ for all $i\leq k$.

Proof.

Part (2) implies part (1) by Lemma 3.4(1).

We now need to show that (1) implies (2). Suppose that $[u^{\prime}]\in S$. Then $||u^{\prime}||_{A}\leq(1+\varepsilon)||u||_{A}$. Since ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$, there is $\varphi\in{\rm Out}(F_{N})$ such that $\varphi[u]=[u^{\prime}]$. Proposition 2.5 implies that there exist $\tau_{1},\dots,\tau_{k}\in\mathcal{W}_{N}$ such that for $[u_{0}]=[u]$, $[u_{i}]=\tau_{i}\dots\tau_{1}([u])$ for $i=1,\dots,k$ we have

[TABLE]

and that $[u_{k}]=[u^{\prime}]$. We can assume that we have eliminated repetitions among $[u_{i}]$, so that $[u]=[u_{0}],[u_{1}],\dots,[u_{k}]=[u^{\prime}]$ are distinct.

Case 1. Suppose first that for every $i\geq 1$ we have $||u_{i}||_{A}\leq(1+\varepsilon)||u_{i-1}||_{A}$. Since $[u_{0}]=[u]\in S$, Lemma 3.4(1) then implies that $[u_{i}]\in S$ for all $i=1,\dots,k$. Since $\#S\leq M$ and all $[u_{i}]$ are distinct, it follows that $k\leq M$. Thus the conclusion of part (2) of the lemma holds in this case.

Case 2. Suppose there is some $i\geq 1$ we have $||u_{i}||_{A}>(1+\varepsilon)||u_{i-1}||_{A}$. Let $i_{0}$ be the smallest among such $i$. Then for all $j<i_{0}$ we have $||u_{j}||_{A}\leq(1+\varepsilon)||u_{j-1}||_{A}$, and we also have $||u_{i_{0}}||_{A}>(1+\varepsilon)||u_{i_{0}-1}||_{A}$. Again by Lemma 3.4(1) we conclude that $[u_{j}]\in S$ for all $j<i_{0}$. In particular $[u_{i_{0}-1}]\in S$. Since for $[u_{i_{0}}]=\tau_{i_{0}}[u_{i_{0}-1}]$ we have $||u_{i_{0}}||_{A}>(1+\varepsilon)||u_{i_{0}-1}||_{A}$, condition (3) of Definition 3.3 implies that $[u_{i_{0}}]\not\in S$. Therefore by part (4)(ii) of Definition 3.3 we have $||u_{i_{0}}||_{A}\geq\lambda||u_{i_{0}-1}||_{A}$. Since $[u],[u_{i_{0}-1}]\in S$, we have $||u_{i_{0}}||_{A}\geq(1-\varepsilon)||u||_{A}$. Therefore $||u_{i_{0}}||_{A}\geq\lambda(1-\varepsilon)||u||_{A}>(1+\varepsilon)||u||_{A}$, yielding a contradiction. Thus Case 2 is impossible.

Therefore the conclusion of part (2) of the lemma holds, as required.

∎

Corollary 3.17**.**

Let $M\geq 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be rational numbers such that $\varepsilon<\lambda-1$ and $\lambda\frac{1-\varepsilon}{1+\varepsilon}>1$. Then there is an algorithm that, given $1\neq[u]\in\mathcal{C}_{N}$ decides in linear time in $||u||_{A}$ whether or not $[u]$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal (that is, whether $[u]$ belongs to some $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing set $S$).

Proof.

Suppose we are given an input $1\neq[u]\in\mathcal{C}_{N}$. We need to decide if there exists an $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing set $S$ containing $[u]$.

We first enumerate all chains of $k\leq M$ Whitehead moves as in part (2) of Lemma 3.16 and collect all $[u^{\prime}]$ reachable from $[u]$ by applying such chains. Denote the resulting subset of $\mathcal{C}_{N}$ by $S^{\prime}$. Computing $S^{\prime}$ from $[u]$ takes at most linear time in $||u||_{A}$ since $M$ is fixed and the set $\mathcal{W}_{N}$ is also finite and fixed.

Lemma 3.16 implies that if $[u]$ belongs to some $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing set $S$ then $S=S^{\prime}$. We then check if conditions (1)-(4) of Definition 3.3 hold for $S^{\prime}$. Again this can be done in linear time in $||u||_{A}$ since $M$ is fixed.

We conclude that $[u]$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimal if and only if conditions (1)-(4) of Definition 3.3 do hold for $S^{\prime}$. ∎

4. Geodesic currents on free groups

We provide some basic background on geodesic currents on $F_{N}$ here and refer the reader to [23, 26, 27] for further details. For the remainder of this section let $F_{N}$ be a free group of finite rank $N\geq 2$. We denote by $\partial F_{N}$ the hyperbolic boundary of $F_{N}$ and denote $\partial^{2}F_{N}:=\{(x,y)|x,y\in\partial F_{N},x\neq y\}$. We give $\partial^{2}F_{N}$ the subspace topology from $\partial F_{N}\times\partial F_{N}$ and endow $\partial^{2}F_{N}$ with the natural diagonal translation action of $F_{N}$ by homeomorphisms. The space $\partial^{2}F_{N}$ also comes with a natural “flip” involution $\varpi:\partial^{2}F_{N}\to\partial^{2}F_{N}$, $\varpi:(x,y)\mapsto(y,x)$. The boundary $\partial F_{N}$ is homeomorphic to the Cantor set, and $\partial^{2}F_{N}$ is a locally compact totally disconnected but non-compact metrizable topological space.

4.1. Basic notions

Definition 4.1**.**

A geodesic current on $F_{N}$ is a locally finite (i.e. finite on compact subsets) positive Borel measure $\nu$ on $\partial^{2}F_{N}$ such that $\nu$ is $F_{N}$-invariant and flip-invariant. The set of all geodesic currents on $F_{N}$ is denoted $\mbox{Curr}(F_{N})$.

The set $\mbox{Curr}(F_{N})$ is equipped with the weak-* topology, which makes $\mbox{Curr}(F_{N})$ locally compact. Any automorphism $\Phi\in{\rm Out}(F_{N})$ is a quasi-isometry of $F_{N}$ and hence extends to a homeomorphism, which we still denote by $\Phi:\partial F_{N}\to\partial F_{N}$. Diagonally extending this homeomorphism we also get a homeomorphism $\Phi:\partial^{2}F_{N}\to\partial^{2}F_{N}$. There is a natural left action of ${\rm Aut}(F_{N})$ by homeomorphisms on $\mbox{Curr}(F_{N})$, where for $\Phi\in{\rm Aut}(F_{N})$ and $\nu\in\mbox{Curr}(F_{N})$ we have $(\Phi\nu)(S)=\nu(\Phi^{-1}(S))$ for $S\subseteq\partial^{2}F_{N}$. The subgroup ${\rm Inn}(F_{N})\leq{\rm Aut}(F_{N})$ is contained in the kernel of this action, and therefore the action descends to the action of ${\rm Out}(F_{N})$ on $\mbox{Curr}(F_{N})$. There is also a multiplication by a scalar action of $\mathbb{R}_{>0}$ in $\mbox{Curr}(F_{N})-\{0\}$, with the quotient space $\mathbb{P}\mbox{Curr}(F_{N})=(\mbox{Curr}(F_{N})-\{0\})/\mathbb{R}_{>0}$, equipped with the quotient topology. The space $\mathbb{P}\mbox{Curr}(F_{N})$ is compact, although infinite dimensional. For $0\neq\nu\in\mbox{Curr}(F_{N})$ we denote the $\mathbb{R}_{>0}$-equivalence class of $\nu$ by $[\nu]$. Thus $[\nu]=\{c\nu|c\in\mathbb{R}_{>0}\}$ and $[\nu]\in\mathbb{P}\mbox{Curr}(F_{N})$. We call elements of $\mathbb{P}\mbox{Curr}(F_{N})$ projectivized geodesic currents on $F_{N}$.

Let $1\neq g\in F_{N}$. Then $g$ determines a pair of distinct “poles” $g^{-\infty},g^{\infty}\in\partial F_{N}$, where $g^{\infty}=\lim_{n\to\infty}g^{n}$ and $g^{-\infty}=\lim_{n\to\infty}g^{-n}$ in $F_{N}\cup\partial F_{N}$. Thus $(g^{-\infty},g^{\infty})\in\partial^{2}F_{N}$. For $h\in F_{N}$ we have $hg^{\infty}=(hgh^{-1})^{\infty}$, and we also have $g^{-\infty}=(g^{-1})^{\infty}$.

Definition 4.2** (Counting and rational currents).**

Let $1\neq g\in F_{N}$. Then

[TABLE]

is a geodesic current on $F_{N}$ called the counting current for $g$. We call currents of the form $c\eta_{g}\in\mbox{Curr}(F_{N})$, where $c>0$ and $1\neq g\in F_{N}$, rational currents.

It is known that the set of all rational currents is a dense subset of $\mbox{Curr}(F_{N})$, and that for any $1\neq g\in F_{N}$ and any $u\in F_{N}$ we have $\eta_{g}=\eta_{ugu^{-1}}=\eta_{g^{-1}}$. Therefore we also denote $\eta_{[g]}:=\eta_{g}$ where $[g]$ is the conjugacy class of $g$ in $F_{N}$. Moreover, for $\varphi\in{\rm Aut}(F_{N})$ and $1\neq g\in F_{N}$, one has $\varphi\eta_{g}=\eta_{\varphi(g)}$.

4.2. Simplicial charts and weights

We adopt the conventions of [14] regarding graphs. All graphs are 1-cell complexes, where 0-cells are called vertices and 1-cells are called topological edges. Every topological edge is homeomorphic to an interval $(0,1)$ and thus admits exactly two orientations. An oriented edge of a $e$ graph is a topological edge with a choice of an orientation. The same topological edge with the opposite orientation is denoted $e^{-1}$. The set of all oriented edges of a graph $\Delta$ is denoted $E\Delta$. We also denote by $V\Delta$ the set of all vertices of $\Delta$. Unless specified otherwise, by an edge of a graph we always mean an oriented edge. Every oriented edge $e\in E\Delta$ has an initial vertex denoted $o(e)\in V\Delta$ and a terminal vertex $t(e)\in E\Delta$. We also have $o(e^{-1})=t(e)$ and $t(e^{-1})=o(e)$. An edge-path $\gamma$ of length $n\geq 1$ in $\Delta$ is a sequence of edges $e_{1},\dots,e_{n}$ such that $t(e_{i})=o(e_{i+1})$. We also consider a vertex $v$ of $\Delta$ to be a path of length [math]. An edge-path $\gamma$ in $\Delta$ is reduced if it does not contain subpaths of the form $e,e^{-1}$ where $e\in E\Delta$. We denote by $|\gamma|$ the length of an edge-path $\gamma$.

Definition 4.3** (Simplicial chart).**

Let $F_{N}$ be free of rank $N\geq 2$. A simplicial chart on $F_{N}$ is a pair $(\Gamma,\kappa)$ where $\Gamma$ is a finite connected oriented graph with all vertices of degree $\geq 3$ and with the first betti number $b(\Gamma)=N$, and that where $\kappa:F_{N}\to\pi_{1}(\Gamma,x_{0})$ is a group isomorphism (with $x_{0}\in V\Gamma$ some base-vertex), called a marking.

When talking about simplicial charts, we usually suppress explicit mention of $\kappa$. We equip $\Gamma$ and $T_{0}=\widetilde{(\Gamma,x_{0})}$ with simplicial metrics, where every edge has length $1$. In this setting we denote by $\Omega(\Gamma)$ the set of all semi-infinite reduced edge-paths $e_{1},e_{2},\dots,$ in $\Gamma$. For $n\geq 1$ denote by $\Omega_{n}(\Gamma)$ the set of all reduced edge-paths $e_{1},e_{2},\dots,e_{n}$ of length $n$ in $\Gamma$. Also denote $\Omega_{\ast}=\cup_{n=1}^{\infty}\Omega_{n}(\Gamma)$.

If $A=\{a_{1},\dots,a_{N}\}$ is a free basis of $F_{N}$, then the graph $R_{A}$, with a single vertex $x_{0}$ and with $N$ petal-edges marked $a_{1},\dots,a_{N}$, is a simplicial chart on $F_{N}$. In this case the corresponding covering tree $T_{A}:=\widetilde{R}_{A}$ is exactly the Cayley tree of $F_{N}$ with respect to $A$. We refer to such simplicial chart $R_{A}$ as an $N$-rose.

For a simplicial chart $\Gamma$, the marking $\kappa$ induces an $F_{N}$-equivariant quasi-isometry $F_{N}\to T_{0}$, which we use to identify $\partial F_{N}$ with $\partial T_{0}$. For $(x,y)\in\partial^{2}F_{N}$ denote by $\gamma_{x,y}$ the bi-infinite geodesic in $T_{0}$ from $x$ to $y$. The group $F_{N}=\pi_{1}(\Gamma,x_{0})$ acts on $T_{0}=\widetilde{\Gamma}$ by covering transformations, which is a free and isometric discrete action with $T_{0}/F_{N}=\Gamma$.

Definition 4.4** (Cylinders and weights).**

Let $\Gamma$ be a simplicial chart on $F_{N}$, with $T_{0}=\widetilde{\Gamma}$.

(1) For two distinct vertices $p,q\in T_{0}$ denote by $Cyl_{\Gamma}([p,q])$ the set of all $(x,y)\in\partial^{2}F_{N}$ such that the bi-infinite geodesic $\gamma_{x,y}$ contains $[p,q]$ as a subsegment. The set $Cyl_{\Gamma}([p,q])\subseteq\partial^{2}F_{N}$ is called the cylinder set corresponding to $[p,q]$.

For any $g\in F_{N}$ and any $p,q\in VT_{0},p\neq q$ we have $gCyl_{\Gamma}([p,q])=Cyl_{\Gamma}([gp,gq])$. The cylinder sert $Cyl_{\Gamma}([p,q])\subseteq\partial^{2}F_{N}$ are compact and open, and the collection of all such cylinder sets forms a basis for the subspace topology on $\partial^{2}F_{N}$ defined above.

(2) For a geodesic current $\eta\in\mbox{Curr}(F_{N})$ denote by $\langle v,\eta\rangle_{\Gamma}:=\eta\left(Cyl_{\Gamma}([p,q])\right)$ where $[p,q]$ is any lift of $v$ to $T_{0}$. The number $0\leq\langle v,\eta\rangle_{\Gamma}<\infty$ is called the weight of $v$ in $\eta$ with respect to $\Gamma$.

If $\Gamma=R_{A}$ is an $N$-rose, we use the subscript $A$ rather than $R_{A}$ for chart-related notations. E.g. $\langle v,\eta\rangle_{A}:=\langle v,\eta\rangle_{R_{A}}$, etc.

Proposition 4.5**.**

[23]** Let $F_{N}$ be free of rank $N\geq 2$ and let $\Gamma$ be a simplicial chart on $F_{N}$. Then:

1. (1)

For $\eta,\eta_{n}\in\mbox{Curr}(F_{N})$, where $n=1,2,\dots$, we have $\lim_{n\to\infty}\eta_{n}=\eta$ in $\mbox{Curr}(F_{N})$ if and only if for every $v\in\Omega_{\ast}(\Gamma)$ we have

[TABLE] 2. (2)

Let $\eta\in\mbox{Curr}(F_{N})$. Then for every $k\geq 1$ and every $v\in\Omega_{k}(\Gamma)$ we have

[TABLE]

Moreover, any system of finite nonnegative weights on $\Omega_{\ast}(\Gamma)$ satisfying uniquely determines a current $\eta\in\mbox{Curr}(F_{N})$ realizing these weights.

Condition $({\ddagger})$ is often called the switch condition for $\Gamma$.

For $v\in\Omega_{\ast}(\Gamma)$ and a nondegenerate closed reduced and cyclically reduced edge-path $w$ in $\Gamma$, denote by $\langle v,w\rangle_{\Gamma}$ the number of ways in which $v$ can be read, reading forwards or backwards, in a circle of length $|w|$ labelled by $w$. The number $\langle v,w\rangle_{\Gamma}\geq 0$ is called the number of occurrences of $v$ in $w$. A key useful fact that follows from the definitions is:

Lemma 4.6**.**

Let $F_{N}$ be free of rank $N\geq 2$ and let $\Gamma$ be a simplicial chart on $F_{N}$. Let $v\in\Omega_{\ast}(\Gamma)$ and let $w$ be a nondegenerate closed reduced and cyclically reduced edge-path in $\Gamma$. Then $\langle v,w\rangle_{\Gamma}=\langle v,\eta_{w}\rangle_{\Gamma}$.

∎

Definition 4.7** (Uniform current).**

Let $F_{N}=F(A)$ be free of rank $N\geq 2$ with a free basis $A$. The uniform current $\nu_{A}\in\mbox{Curr}(F_{N})$ corresponding to $A$ is the current given by the weights $\langle v,\nu_{A}\rangle_{A}=\frac{1}{N(2N-1)^{k-1}}$ for every $1\neq v\in F_{N}$ with $|v|_{A}=k\geq 1$.

For a current $\eta\in\mbox{Curr}(F_{N})$ the support $\mbox{Supp}(\eta)\subseteq\partial^{2}F_{N}$ is

[TABLE]

Thus $\mbox{Supp}(\eta)$ is a closed $F_{N}$-invariant subset of $\partial^{2}F_{N}$.

Remark 4.8*.*

Let $\Gamma$ be a simplicial chart on $F_{N}$. If $\eta\in\mbox{Curr}(F_{N})$ and $(x,y)\in\partial^{2}F_{N}$ then $(x,y)\in\mbox{Supp}(\mu)$ if and only if every finite nondegenerate edge subpath of $\gamma_{x,y}$ projects to a reduced edge-path $v$ in $\Gamma$ with $\langle v,\eta\rangle_{\Gamma}>0$.

4.3. Geometric intersection form

We refer the reader to [3, 16, 27, 45] for the background and basic info regarding the Outer space, and only recall a few facts and definitions here. Denote by $\mbox{cv}_{N}$ the (unprojectivized) Culler-Vogtmann Outer space for $F_{N}$. Elements of $\mbox{cv}_{N}$ are equivariant $F_{N}$-isometry classes of free and discrete minimal isometric actions of $F_{N}$ on $\mathbb{R}$-trees. In particular, if $\Gamma$ is a simplicial chart on $F_{N}$ then $T_{0}=\widetilde{\Gamma}$ defines a point of $\mbox{cv}_{N}$. There is a natural “axes” topology on $\mbox{cv}_{N}$ and a (right) action of ${\rm Out}(F_{N})$ on $\mbox{cv}_{N}$ by homeomorphisms. Moreover, the closure $\overline{\mbox{cv}}_{N}$ of $\mbox{cv}_{N}$ in the axes topology is known to consist of all minimal nontrivial ”very small” isometric actions on $F_{N}$ on $\mathbb{R}$-trees (again considered up to $F_{N}$-equivariant isometry), and the action of ${\rm Out}(F_{N})$ extends to $\overline{\mbox{cv}}_{N}$. For $T\in\overline{\mbox{cv}}_{N}$ and $g\in F_{N}$ denote by $||g||_{T}$ the translation length of $g$ in $T$, that is $||g||_{T}=\inf_{x\in T}d_{T}(x,gx)$.

A key result of Kapovich and Listing [26] is:

Proposition 4.9**.**

Let $F_{N}$ be free of finite rank $N\geq 2$. Then there exists a continuous geometric intersection form

[TABLE]

satisfying the following properties:

1. (1)

The map $\langle-\,,\,-\rangle$ is $\mathbb{R}_{\geq 0}$-homogeneous with respect to the first argument and $\mathbb{R}_{\geq 0}$-linear with respect to the second argument. 2. (2)

For every $\varphi\in{\rm Out}(F_{N})$, every $T\in\overline{\mbox{cv}}_{N}$ and every $\eta\in\mbox{Curr}(F_{N})$ we have

[TABLE] 3. (3)

For every $1\neq g\in F_{N}$ and every $T\in\overline{\mbox{cv}}_{N}$ we have $\langle T,\eta_{g}\rangle=||g||_{T}$.

In view of the above proposition, for $T\in\overline{\mbox{cv}}_{N}$ and $\eta\in\mbox{Curr}(F_{N})$ we denote $||\eta||_{T}=\langle T,\eta\rangle$.

For every $T\in\overline{\mbox{cv}}_{N}$ there is an associated dual lamination $L(T)\subseteq\partial^{2}F_{N}$, which is a certain closed $F_{N}$-invariant and flip-invariant subset of $\partial^{2}F_{N}$ recording the information about sequences of elements of $F_{N}$ with translation length in $T$ converging to [math]. We refer the reader to [27] for the precise definition of $L(T)$ and additional details.

We need the following key result of [27]:

Proposition 4.10**.**

Let $T\in\overline{\mbox{cv}}_{N}$ and $\eta\in\mbox{Curr}(F_{N})$. Then $||\eta||_{T}=0$ if and only if $\mbox{Supp}(\eta)\subseteq L(T)$.

5. Filling geodesic currents

Definition 5.1**.**

Let $F_{N}$ be free of rank $N\geq 2$.

1. (1)

An element $g\in F_{N}$ is filling in $F_{N}$ if for every $T\in\overline{\mbox{cv}}_{N}$ we have $||g||_{T}>0$. 2. (2)

A current $\eta\in\mbox{Curr}(F_{N})$ is filling in $F_{N}$ if for every $T\in\overline{\mbox{cv}}_{N}$ we have $||\eta||_{T}>0$.

Thus an element $1\neq g\in F_{N}$ is filling if and only if $\eta_{g}$ is a filling.

One of the main results of [27] is:

Proposition 5.2**.**

[27, Corollary 1.6]** Let $\eta\in\mbox{Curr}(F_{N})$ be such that $\mbox{Supp}(\eta)=\partial^{2}F_{N}$. Then $\eta$ is filling in $F_{N}$.

We will sometimes say that a current $\eta\in\mbox{Curr}(F_{N})$ has full support if $\mbox{Supp}(\eta)=\partial^{2}F_{N}$.

Remark 4.8 directly implies:

Proposition 5.3**.**

Let $\Gamma$ be a simplicial chart on $F_{N}$.

Then $\eta\in\mbox{Curr}(F_{N})$ has full support if and only if for every nondegenerate edge-path $v$ in $\Gamma$ we have $\langle v,\eta\rangle_{\Gamma}>0$.

Lemma 5.4**.**

Let $0\neq\nu\in\mbox{Curr}(F_{N})$. Let $w$ be a nondegenerated closed reduced and cyclically reduced edge-path in $\Gamma$ such that for every $n\geq 1$ we have $\langle w^{n},\nu\rangle_{\Gamma}>0$. Then $\mbox{Supp}(\eta_{w})\subseteq\mbox{Supp}(\nu)$.

Proof.

Since geodesic currents are flip-invariant, the assumptions of the lemma imply that the points $p_{+}=(w^{-\infty},w^{\infty}),p_{-}=(w^{\infty},w^{-\infty})\in\partial^{2}F_{N}$ belong to $\mbox{Supp}(\nu)$. Since $\mbox{Supp}(\eta_{w})=\cup_{h\in F_{N}}h\{p_{+},p_{-}\}$, we then have $\mbox{Supp}(\eta_{w})\subseteq\mbox{Supp}(\nu)$. ∎

Lemma 5.5**.**

Let $1\neq g\in F_{N}$ be a filling element and let $0\neq\nu\in\mbox{Curr}(F_{N})$ be a current such that $\mbox{Supp}(\eta_{g})\subseteq\mbox{Supp}(\nu)$. Then $\nu$ is a filling current.

Proof.

Suppose, on the contrary, that $\nu$ is not filling. Then there exists $T\in\mbox{cv}_{N}$ such that $\langle T,\nu\rangle=0$. By [27, Theorem 1.1] this implies that $\mbox{Supp}(\nu)\subseteq L(T)$. Hence $\mbox{Supp}(\eta_{g})\subseteq L(T)$ as well. Therefore, again by [27, Theorem 1.1], we have $0=\langle T,\eta_{g}\rangle=||g||_{T}$, which contradicts that $g$ is filling. ∎

Corollary 5.6**.**

Let $z$ be a nondegenerated closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of a filling element $g\in F_{N}$.

Let $0\neq\nu\in\mbox{Curr}(F_{N})$ be such that for every $n\geq 1$ we have $\langle z^{n},\nu\rangle_{\Gamma}>0$. Then $\nu$ is a filling current.

Proof.

Lemma 5.4 implies that $\mbox{Supp}(\eta_{g})\subseteq\mbox{Supp}(\nu)$. Therefore, by Lemma 5.5, the current $\nu$ is filling. ∎

Proposition 5.7**.**

Let $0\neq\nu\in\mbox{Curr}(F_{N})$ be such that for some free basis $A=\{a_{1},\dots,a_{N}\}$ the following holds. For $i=1,\dots,N$ let $w_{i}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_{i}$ in $F_{N}$. For $1\leq i<j\leq N$ let $w_{i,j}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_{i}$ in $F_{N}$. Suppose that we have $\langle w_{i}^{n},\nu\rangle_{\Gamma}>0$ for $i=1,\dots,N$ and that we have $\langle w_{ij}^{n},\nu\rangle_{\Gamma}>0$ for all $1\leq i<j\leq N$ and all $n\geq 1$. Then the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling.

Proof.

Indeed, suppose $\nu$ is not filling. Then there exists $T\in\overline{\mbox{cv}}_{N}$ such that $\langle T,\nu\rangle=0$. By [27, Theorem 1.1] this implies that $\mbox{Supp}(\nu)\subseteq L(T)$.

Lemma 5.4 implies that for all $i=1,\dots,N$ we have $\mbox{Supp}(\eta_{a_{i}})\subseteq\mbox{Supp}(\nu)$, and for all $1\leq i<j\leq N$ we have $\mbox{Supp}(\eta_{a_{i}a_{j}})\subseteq\mbox{Supp}(\nu)$. Since $\mbox{Supp}(\nu)\subseteq L(T)$, [27, Theorem 1.1] implies that for $i=1,\dots,N$

[TABLE]

and for all $1\leq i<j\leq N$ we have

[TABLE]

Thus all $a_{i}$ and $a_{i}a_{j}$ act elliptically on $T$ and so have nonempty fixed sets in $T$.

For $1\leq i<j\leq N$, the elements $a_{i},a_{j},a_{ij}$ act elliptically on $T$, and therefore, by [43, Proposition 1.8], $Fix_{T}(a_{i})\cap Fix_{T}(a_{j})\neq\varnothing$. Thus $Fix_{T}(a_{1}),\dots Fix_{T}(a_{N})$ are nonempty subtrees of $T$ with pairwise nonempty intersections. Therefore $\cap_{i=1}^{N}Fix_{T}(a_{i})\neq\varnothing$. Hence $F_{N}$ has a global fixed point in $T$, which contradicts the fact that $T\in\overline{\mbox{cv}}_{N}$ is a nontrivial $F_{N}$-tree. ∎

Proposition 5.8**.**

Let $F_{N}=F(A)$ (where $N\geq 2$) and let $w\in F(A)$ be a freely and cyclically reduced word such that for every $v\in F(A)$ with $|v|_{A}=3$, the word $v$ occurs as a subword of some cyclic permutation of $w$ or of $w^{-1}$. Then:

1. (1)

The element $w\in F_{N}$ is filling. 2. (2)

If $0\neq\nu\in\mbox{Curr}(F_{N})$ is such that for all $n\geq 1$ $\langle w^{n},\nu\rangle_{A}>0$, then the current $\nu$ is filling in $F_{N}$.

Proof.

Part (1) is exactly [7, Corollary 5.6].

Now [art (1) implies part (2) by Corollary 5.6. ∎

6. Filling currents and $(M,\lambda,\varepsilon)$-minimality

Also, as before, we denote by $T_{A}$ the Cayley graph of $F_{N}$ with respect to the free basis $A$. Thus $T_{A}$ is a simplicial tree with all edges of length $1$.

Definition 6.1**.**

Let $0\neq\nu\in Curr(F_{N})$. The automorphic distortion spectrum of $\nu$ with respect to the free basis $A$ of $F_{N}$ is the set

[TABLE]

Also denote $J_{A}(\nu):=\inf D_{A}(\nu)$.

Remark 6.2*.*

Thus $D_{A}(\nu)\subseteq\mathbb{R}_{>0}$. Since $\langle T_{A},\varphi\nu,\rangle=\langle T_{A}\varphi,\nu\rangle=\langle T_{\varphi(A)},\nu\rangle$, it is easy to see that the set $D_{A}(\nu)$ is independent of the choice of a free basis $A$ of $F_{N}$ and depends only on $\nu$. Nevertheless, we will keep the subscript $A$ in the notation $D_{A}(\nu)$ since for our purposes the fixed choice of $A$ is important.

Note also that for $1\neq w\in F_{N}$ and $\varphi\in{\rm Out}(F_{N})$ we have $\langle T_{A},\varphi\eta_{w}\rangle=\langle T_{A},\eta_{\varphi}(w)\rangle=||\varphi(w)||_{A}$. Therefore in this case $D_{A}(\eta_{w})=\{||\varphi(w)||_{A}|\varphi\in{\rm Out}(F_{N})\}\subseteq\mathbb{Z}_{>0}$, and $J_{A}(\eta_{w})$ is the smallest $||.||_{A}$-length of elements in the orbit ${\rm Out}(F_{N})[w]$.

We need the following useful result essentially proved in [27, Theorem 1.2]:

Proposition 6.3**.**

Let $\nu\in\mbox{Curr}(F_{N})$ (where $N\geq 2$) be a filling current and let $A$ be a free basis of $F_{N}$. Then:

1. (1)

The set $D_{A}(\nu)$ is a discrete unbounded subset of $[0,\infty)$. 2. (2)

For every $C>0$ the set $\{\varphi\in{\rm Out}(F_{N})|\langle T_{A},\varphi\nu\rangle\leq C\}$ is finite.

Proof.

The proof is a verbatim copy of the proof of [27, Theorem 11.2] where the same result was established under the assumption that $\nu\in\mbox{Curr}(F_{N})$ is filling. The only place in the proof of Theorem 11.2 in [27] where the filling assumption on $\nu$ was used is at the bottom of page 1461 in [27], to show that $\langle T_{\infty},\nu\rangle\neq 0$ for a certain tree $T_{\infty}\in\overline{\mbox{cv}}_{N}$ constructed earlier in the proof. However, in our case $\langle T_{\infty},\nu\rangle\neq 0$ since $\nu$ is assumed to be filling in the present proposition. ∎

Proposition 6.3 immediately implies:

Corollary 6.4**.**

For $F_{N}$ and $A$ as in Proposition 6.3 let $\nu\in Curr(F_{N})$ be a filling current. Then:

1. (1)

We have $J_{A}(\nu)\in D_{A}(\nu)$, so that $J_{A}(\nu)=\min D_{A}(\nu)$. 2. (2)

The set $\Delta_{A}(\nu)=\{\varphi\in{\rm Out}(F_{N})|\langle T_{A},\varphi\nu\rangle=J_{A}(\nu)\}$ is finite and nonempty.

Definition 6.5**.**

Let $F_{N}$ be free of finite rank $N\geq 2$, let $A$ be a free basis of $A$ and let $\nu\in Curr(F_{N})$ be a filling current. We call the set $\Delta_{A}(\nu):=\{\varphi\in{\rm Out}(F_{N})|\langle T_{A},\varphi\nu\rangle=J_{A}(\nu)\}$ the $A$-minimizing set for $\nu$ and we call the integer $M_{A}(\nu):=\#\Delta_{A}(\nu)\geq 1$ the minimizing multiplicity for $\nu$ with respect to $A$. Also put

Also let $J^{\prime}_{A}(\nu)=\min(D_{A}(\nu)\setminus\{J_{A}(\nu)\})$ and let $\lambda_{A}(\nu)=\frac{J^{\prime}_{A}(\nu)}{J_{A}(\nu)}$, so that $\lambda_{A}(\nu)>1$. We call $\lambda_{A}(\nu)$ the distortion threshold for $\nu$ with respect to $A$. Finally denote $\Im_{A}(\nu)=\Delta_{A}(\nu)\nu=\{\varphi\nu|\varphi\in\Delta_{A}(\nu)\}\subseteq Curr(F_{N})$ and call $\Im_{A}$ the orbit floor for $\nu$.

The following statement is a key technical result of this paper:

Theorem 6.6**.**

Let $F_{N}$ be free of finite rank $N\geq 2$, let $A$ be a free basis of $A$ and let $\nu\in\mbox{Curr}(F_{N})$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_{A}(\nu)$ and let $0<\varepsilon<1$ be such that $\lambda_{A}(\nu)>\lambda>1+\varepsilon$.

Let $\mathfrak{W}=\Delta_{A}(\nu)$ and let $M=M_{A}(\nu)=\#\mathfrak{W}$.

Then there exists a neighborhood $U=U([\nu],\lambda,\varepsilon)$ of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ such that for every $1\neq w\in F_{N}$ with $[\eta_{w}]\in U$ the set $S=\mathfrak{W}[w]\subseteq\mathcal{C}_{N}$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing.

Proof.

Denote $\Im=\Im_{A}(\nu)=\mathfrak{W}\nu\subseteq\mbox{Curr}(F_{N})$. Since $\Im=\mathfrak{W}\nu$ and $S=\mathfrak{W}[w]$, it follows that $\#\Im\leq M$ and $\#S\leq M$. Also, by construction, $S\subseteq{\rm Out}(F_{N})[w]$. Therefore for every $[u],[u^{\prime}]\in S$ we have ${\rm Out}(F_{N})[u]={\rm Out}(F_{N})[u^{\prime}]$. Thus conditions (1) and (2) of Definition 3.3 hold for $S$.

We also have $||\nu^{\prime}||_{A}=J_{A}(\nu)$ for all $\nu^{\prime}\in\Im$. Moreover, if $\nu^{\prime}\in\Im$ and $\psi\in{\rm Out}(F_{N})$ is such that $\psi\nu^{\prime}\not\in\Im$ then $||\psi\nu^{\prime}||_{A}/J_{A}(\nu)\geq\lambda_{A}(\nu)>1$. In particular, the latter statement holds whenever $\tau\in\mathcal{W}_{N}$ is a Whitehead move such that $\tau\nu^{\prime}\not\in\Im$. For each $\nu^{\prime}\in\Im$ denote

[TABLE]

Proposition 6.3 also implies that for each $\nu^{\prime}\in\Im$ the set $R_{\Im}(\nu^{\prime})$ is finite.

Moreover, for every $\nu^{\prime},\nu^{\prime\prime}\in\Im$ there are $\varphi^{\prime},\varphi^{\prime\prime}\in\mathfrak{W}$ such that $\varphi^{\prime}\nu=\nu^{\prime}$ and $\varphi^{\prime\prime}\nu=\nu^{\prime\prime}$ so that $\varphi^{\prime\prime}(\varphi^{\prime})^{-1}\in R_{\Im}(\nu^{\prime})$ and $\varphi^{\prime}(\varphi^{\prime\prime})^{-1}\in R_{\Im}(\nu^{\prime\prime})$. Therefore for every $\nu^{\prime}\in\Im$ we have $R_{\Im}(\nu^{\prime})\nu^{\prime}=\Im$. For exactly the same reason, if $[u]=\varphi^{\prime}[w]$, where $\varphi^{\prime}\in\mathfrak{W}$ and $\nu^{\prime}=\varphi^{\prime}\nu\in\Im$ then

[TABLE]

Since $\lambda_{A}(\nu)(1-2\varepsilon)>\lambda>1+\varepsilon$, we can choose $\lambda<\lambda_{1}<\lambda_{A}(\nu)$ so that

[TABLE]

By continuity of the intersection form $\langle-,-\rangle$ and of the action of ${\rm Out}(F_{N})$ on $\mathbb{P}Curr(F_{N})$, there exist neighborhoods $U([\nu^{\prime}])$ of $[\nu^{\prime}]$ in $\mathbb{P}Curr(F_{N})$, where $\nu^{\prime}\in S$, and there exists a neighborhood $U([\nu])$ of $[\nu]$, such that the following hold:

• (a)

If $\nu^{\prime}\in\Im$ and $[\eta]\in U([\nu^{\prime}])$ then for every $\psi\in R_{\Im}(\nu^{\prime})$ with $\nu^{\prime\prime}=\psi\nu\in\Im$ we have $\psi[\eta]\in U([\nu^{\prime\prime}])$ and

[TABLE]

• (b)

If $\nu^{\prime}\in\Im$, $[\eta]\in U([\nu^{\prime}])$ and $\tau\in\mathcal{W}_{N}$ is a Whitehead move such that $\tau\nu^{\prime}\not\in\Im$ then

[TABLE]

• (c)

For every $\varphi\in\mathfrak{W}$ (so that $\psi\nu\in\Im$) we have $\varphi U([\nu])\subseteq U([\varphi\nu])$.

Recall that $S=\mathfrak{W}[w]=\{\varphi([w])|\varphi\in\mathfrak{W}\}$.

We will show that for every $1\neq w\in F_{N}$ with $[\eta_{w}]\in U([\nu])$ the set $S$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing, that is that $U([\nu])$ satisfies the conclusion of this proposition.

Thus suppose that $1\neq w\in F_{N}$ is such that $[\eta_{w}]\in U([\nu])$. We have already seen above that conditions (1) and (2) of Definition 3.3 hold for $S$.

Let $[u]\in S$ be arbitrary. Thus $[u]=\varphi^{\prime}[w]$ for some $\varphi\in\mathfrak{W}$, with $\nu^{\prime}=\varphi^{\prime}\nu\in\Im$. By property (c) $\varphi^{\prime}[\eta_{w}]\in U([\nu^{\prime}])$. We also have $\eta_{u}=\varphi^{\prime}\eta_{w}$. Thus $\varphi^{\prime}[\eta_{w}]=[\eta_{u}]\in U([\nu^{\prime}])$.

Claim 0. For any $[x]\in S$ we have $||x||_{A}/||u||_{A}\in[1-\varepsilon,1+\varepsilon]$.

Indeed, let $[x]\in S$, so that $[x]=\varphi^{\prime}[w]$ for some $\psi^{\prime}\in\mathfrak{W}$, so that $\nu^{\prime\prime}=\varphi^{\prime\prime}\nu\in\Im$. Then, as for $[u]$, we have $\varphi^{\prime\prime}[\eta_{w}]=[\eta_{x}]\in U([\nu^{\prime\prime}])$. Then $\psi=\varphi^{\prime\prime}\varphi^{-1}\nu^{\prime}=\nu^{\prime\prime}$, so that $\psi\in R_{\Im}(\nu^{\prime})$. We also have $[x]=\psi[u]$. Then property (a) implies that $||x||_{A}/||u||_{A}\in[1-\varepsilon,1+\varepsilon]$, as required.

Claim 0 shows that conditions (3) of Definition 3.3 hold for $S$.

Claim 1. For any Whitehead move $\tau\in\mathcal{W}_{N}$ exactly one of the following occurs:

(i) We have $||\tau(u)||_{A}/||u||_{A}\in[1-\varepsilon,1+\varepsilon]$, $\tau\in R_{\Im}(\nu^{\prime})$, $\tau\nu^{\prime}\in\Im$ and $\tau[u]\in S$.

(ii) We have $||\tau(u)||_{A}/||u||_{A}\geq\lambda_{1}>\lambda>1+\varepsilon$ and $\tau\not\in R_{\Im}(\nu^{\prime})$ and $\tau[u]\not\in S$.

Indeed, suppose first that $\tau\in\mathbb{R}_{\Im}(\nu^{\prime})$. Thus $\nu^{\prime\prime}=\tau\nu^{\prime}\in\Im$. Since $[\eta_{u}]\in U([\nu^{\prime}])$, property (a) implies that $\tau[\eta_{u}]\in U([\nu^{\prime\prime}])$ and that $||\tau(u)||_{A}/||u||_{A}\in[1-\varepsilon,1+\varepsilon]$. Suppose now that $\tau\not\in\mathbb{R}_{\Im}(\nu^{\prime})$, so that $\tau\nu^{\prime}\not\in\Im$. Since $[\eta_{u}]\in U([\nu^{\prime}])$ property (b) implies that $||\tau(u)||_{A}/||u||_{A}\geq\lambda_{1}>\lambda>1+\varepsilon$. Since $||\tau(u)||_{A}/||u||_{A}>1+\varepsilon$, Claim 0 now implies that $\tau[u]\not\in S$. Thus Claim 1 is verified.

Claim 1 now implies that condition (4) of Definition 3.1 hold for $S$.

Thus $S$ is $(M,\lambda,\varepsilon,\mathcal{W}_{N})$-minimizing, as required. ∎

Corollary 6.7**.**

Let $F_{N}$ be free of finite rank $N\geq 2$, let $A$ be a free basis of $A$ and let $\nu\in\mbox{Curr}(F_{N})$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_{A}(\nu)$ and let $0<\varepsilon<1$ be such that $\lambda_{A}(\nu)>\lambda>1+\varepsilon$.

Let $\mathfrak{W}=\Delta_{A}(\nu)$ and let $M=M_{A}(\nu)=\#\mathfrak{W}$.

Then there exists a neighborhood $U_{1}=U_{1}([\nu],\lambda,\varepsilon)$ of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ such that for every $1\neq w\in F_{N}$ with $[\eta_{w}]\in U$ the set $S=\mathfrak{W}[w]\subseteq\mathcal{C}_{N}$ is $(M,\lambda,\varepsilon)$-minimizing.

Proof.

First choose $\lambda^{\prime}$ such that $\lambda_{A}(\nu)>\lambda^{\prime}>\lambda>1$. Then choose $\varepsilon^{\prime}$ such that $0<\varepsilon^{\prime}<\varepsilon$ and that $\lambda^{\prime}(1-\varepsilon^{\prime})>\lambda$. By Theorem 6.6, there exists a neighborhood $U=U([\nu],\lambda^{\prime},\varepsilon^{\prime})$ of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ such that for every $1\neq w\in F_{N}$ with $[\eta_{w}]\in U$ the set $S=\mathfrak{W}[w]$ is $(M,\lambda^{\prime},\varepsilon^{\prime},\mathcal{W}_{N})$-minimizing. Therefore, by Proposition 3.5, the set $S$ is $(M,\lambda,\varepsilon)$-minimizing. Therefore $U_{1}:=U$ satisfies the requirements of the corollary. ∎

Remark 6.8*.*

Suppose we are in the context of Theorem 6.6 and that $U\subseteq\mathbb{P}\mbox{Curr}(F_{N})$ is a neighborhood of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ provided by the conclusion of Theorem 6.6. Then $U$ contains a “basic” neighborhood $U_{0}\subseteq U$ of $[\nu]$ defined as follows. There exist a finite collection $\mathbb{V}\subseteq F(A)-\{1\}$ and $\varepsilon_{0}>0$ such that for $[\eta]\in\mathbb{P}\mbox{Curr}(F_{N})$ we have $[\eta]\in U_{0}$ if and only if for every $v\in\mathbb{V}$

[TABLE]

Therefore, if $1\neq w\in F_{N}$ is such that for all $v\in\mathbb{V}$

[TABLE]

then $[\eta_{w}]\in U_{0}\subseteq U$ and the conclusion of Theorem 6.6 applies to $w$.

Definition 6.9**.**

Let $\mathcal{W}=W_{1},W_{2},\dots,W_{n},\dots$ be a sequence of $F_{N}$-valued random variables.

1. (1)

We say that $\mathcal{W}$ is tame if for some (equivalently, any) free basis $A$ of $F_{N}$ there exists $C>0$ such that we always have $|W_{n}|_{A}\leq Cn$ where $n\geq 1$. 2. (2)

Let $0\neq\nu\in\mbox{Curr}(F_{N})$. We say that the sequence $\mathcal{W}$ is $\nu$-adapted if a.e. trajectory $w_{1},w_{2},\dots,w_{n},\dots$ of $\mathcal{W}$ we have:

[TABLE]

in $\mathbb{P}Curr(F_{N})$.

In Definition 6.9 above, a random trajectory of $\mathcal{W}$ is implicitly required to satisfy $w_{n}\neq 1$ for all sufficiently large $n$ (which is needed in order for $\eta_{w_{n}}$ to be defined), but we do not require $||w_{n}||_{A}\to\infty$ as $n\to\infty$. In particular, if $\nu=\eta_{w}$ for some $1\neq w\in F_{N}$, and the random process $\mathcal{W}$ always outputs $W_{n}=\eta_{w}$ for all $n\geq 1$, then $\mathcal{W}$ is $\nu$-adapted.

The following statement is key for our paper:

Proposition 6.10**.**

Let $F_{N}$ be free of finite rank $N\geq 2$, let $A$ be a free basis of $A$ and let $\nu\in\mbox{Curr}(F_{N})$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_{A}(\nu)$ and let $\varepsilon>0$ be such that $\lambda_{A}(\nu)>\lambda>1+\varepsilon$.

Let $\mathfrak{W}=\Delta_{A}(\nu)$ and let $M=M_{A}(\nu)=\#\mathfrak{W}$.

Let $\mathcal{W}=W_{1},W_{2},\dots,W_{n},\dots$ be a $\nu$-adapted sequence of $F_{N}$-valued random variables. Then the following hold:

1. (1)

For a.e. trajectory $\xi=(w_{1},w_{2},\dots,w_{n},\dots)$ of $\mathcal{W}$ there exists $n_{0}=n_{0}(\xi)\geq 1$ such that for all $n\geq n_{0}$ the set $S_{n}=\mathfrak{W}[w_{n}]$ is $(M,\lambda,\varepsilon)$-minimizing. 2. (2)

We have

[TABLE]

Proof.

Let $U_{1}=U_{1}([\nu],\lambda,\varepsilon)\subseteq\mathbb{P}\mbox{Curr}(F_{N})$ be a neighborhood of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ whose existence is provided by Corollary 6.7. Choose a basic neighborhood $U_{0}\subseteq U_{1}$ of $[\nu]$ in $\mathbb{P}\mbox{Curr}(F_{N})$ defined by some $\varepsilon>0$ and some finite collection $\mathbb{V}\subseteq F(A)-\{1\}$, as in Remark 6.8

Since $\mathcal{W}$ is adapted to $\nu$, for a.e. trajectory $\xi=(w_{1},w_{2},\dots,w_{n},\dots)$ of $\mathcal{W}$ we have $[\eta_{w_{n}}]\in U_{0}$ and, since a.e. convergence implies convergence in probability, we also have

[TABLE]

Now Corollary 6.7 implies that statements (1) and (2) of Proposition 6.10 hold. ∎

Note in the context of Proposition 6.10, if $1\neq w\in F_{N}$ is such that $S=\mathfrak{W}[w]$ is $(M,\lambda,\varepsilon)$-minimizing then $\#S\leq M$ and every element of $S$ is $(M,\lambda,\varepsilon)$-minimal.

Theorem 6.11**.**

Let $F_{N}=F(A)$ be a free group of finite rank $N\geq 2$ with a free basis $A$.

Let $\mathcal{W}=W_{1},W_{2},\dots$ be a sequence of $F(A)$-valued random variables. Let $0\neq\nu\in\mbox{Curr}(F_{N})$ be a filling geodesic current such that $\mathcal{W}$ is adapted to $\nu$.

Then there exist $M\geq 1$, $\lambda>1$ and a subset $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ with $\#\mathfrak{W}\leq M$ such that for every $0<\varepsilon<1$ with $\lambda>1+\varepsilon$ the following hold:

• (a)

For a.e. trajectory $\xi=(w_{1},w_{2},\dots,w_{n},\dots)$ of $\mathcal{W}$ there exists $n_{0}=n_{0}(\xi)\geq 1$ the following holds for all $n\geq n_{0}$:

1. (1)

The set $S_{n}=\mathfrak{W}[w_{n}]$ is $(M,\lambda,\varepsilon)$-minimizing. 2. (2)

For for every $\varphi\in\mathfrak{W}$ the conjugacy class $\varphi[w_{n}]\in S_{n}$ is $(M,\lambda,\varepsilon)$-minimal. 3. (3)

We have $\mathcal{M}([w_{n}])\subseteq\mathfrak{W}[w_{n}]$, and in particular, $\#\mathcal{M}([w_{n}])\leq M$. 4. (4)

We have ${\rm rank}\ Stab_{{\rm Out}(F_{N})}([w_{n}])\leq K(N,M)$, where $K(N,M)\geq 1$ is some constant depending only on $N$ and $M$.

• (b)

The probability of each of the following events tends to $1$ as $n\to\infty$:

1. (1)

The set $\mathfrak{W}[W_{n}]$ is $(M,\lambda,\varepsilon)$-minimizing. 2. (2)

For for every $\varphi\in\mathfrak{W}$ the conjugacy class $\varphi[W_{n}]$ is $(M,\lambda,\varepsilon)$-minimal. 3. (3)

We have $\mathcal{M}([W_{n}])\subseteq\mathfrak{W}[W_{n}]$, and $\#\mathcal{M}([W_{n}])\leq M$. 4. (4)

We have ${\rm rank}\ Stab_{{\rm Out}(F_{N})}([W_{n}])\leq K(N,M)$, where $K(N,M)\geq 1$ is some constant depending only on $N$ and $M$.

Proof.

Put $\mathfrak{W}=\Delta_{A}(\nu)$ and let $M=M_{A}(\nu)=\#\mathfrak{W}$. Choose $\lambda$ be such that $1<\lambda<\lambda_{A}(\nu)$. L $0<\varepsilon<1$ be such that $\lambda>1+\varepsilon$.

Now Proposition 6.10 implies that statemenst (a)(1) and (b)(1) of Theorem 6.11 holds, which, by definition of $(M,\lambda,\varepsilon)$-minimality, implies that (a)(2) and (b)(2) hold as well. Now Proposition 3.7(1) implies that statements (a)(3) and (b)(3) of Theorem 6.11 holds. Finally, Proposition 3.14 implies that statements (a)(4) and (b)(4) of Theorem 6.11 hold. ∎

Theorem 6.12**.**

Let $F_{N}=F(A)$, $\nu$, $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ and $\mathcal{W}=W_{1},W_{2},\dots$ and be as in Theorem 6.11. Assume also that $\mathcal{W}$ is tame.

Then there exists $K_{0}\geq 1$ such that the following hold:

• (a)

For a.e. independently chosen trajectories $\xi=w_{1},w_{2},\dots$ and $\xi^{\prime}=w_{1}^{\prime},w_{2}^{\prime},\dots$ of $\mathcal{W}$ there exist $n_{0},m_{0}\geq 1$ such that the following hold:

1. (1)

For all $n\geq n_{0}$, the $\mathfrak{W}$-speed-up of Whitehead’s minimization algorithm on the input $w_{n}$ terminates in time at most $K_{0}n$ and produces an element of $\mathcal{M}([w_{n}])$. 2. (2)

If $n\geq n_{0}$, then for any $u\in F_{N}$ the $\mathfrak{W}$-speed-up of Whitehead’s algorithm decides in time at most $K_{0}\max\{n,|u|_{A}^{2}\}$, whether or not ${\rm Aut}(F_{N})w_{n}={\rm Aut}(F_{N})u$. 3. (3)

For all $n\geq n_{0},m\geq m_{0}$, the $\mathfrak{W}$-speed-up of Whitehead’s algorithm decides in time at most $K_{0}\max\{n,m\}$, whether or not ${\rm Aut}(F_{N})w_{n}={\rm Aut}(F_{N})w_{m}^{\prime}$.

• (b)

The probability of each of the following events tends to $1$ as $n\to\infty$:

1. (1)

The $\mathfrak{W}$-speed-up of Whitehead’s minimization algorithm on the input $W_{n}$ terminates in time at most $K_{0}n$ and produces an element of $\mathcal{M}([W_{n}])$. 2. (2)

The element $W_{n}$ has the property that for any $u\in F_{N}$ the $\mathfrak{W}$-speed-up of Whitehead’s algorithm decides in time at most $K_{0}\max\{n,|u|_{A}^{2}\}$, whether or not ${\rm Aut}(F_{N})W_{n}={\rm Aut}(F_{N})u$. 3. (3)

Let $\mathcal{W}^{\prime}=W_{1}^{\prime},W_{2}^{\prime},\dots$ be an independent copy of $\mathcal{W}$. Let $n_{i},m_{i}\geq 1$ be such that $\lim_{i\to\infty}\min\{n_{i},m_{i}\}=\infty$. Then the probability of the following event tends to $1$ as $i\to\infty$:

The $\mathfrak{W}$-speed-up of Whitehead’s algorithm decides in time at most $K_{0}\max\{n_{i},m_{i}\}$, whether or not ${\rm Aut}(F_{N})W_{n_{i}}={\rm Aut}(F_{N})W_{m_{i}}^{\prime}$.

Proof.

Since $\mathcal{W}$ is tame, there exists $C>0$ such that for all $n\geq 1$ we always have $|W_{n}|_{A}\leq Cn$. Let $K\geq 1$ be the constant provided by Theorem 3.11.

We will show that part (a) of Theorem 6.12 holds as the proof of part (b) is essentially the same.

By part (1) of Theorem 6.11 we know that for all big enough $n\geq n_{0}$ (where $n_{0}$ depends on the random trajectory $\xi$) the set $S_{n}=\mathfrak{W}[w_{n}]$ is $(M,\lambda,\varepsilon)$-minimizing and every element of $S_{n}$ is $(M,\lambda,\varepsilon)$-minimal. The same is true for all $S_{m}^{\prime}=\mathfrak{W}[w_{m}^{\prime}]$ for all $m\geq m_{0}=m_{0}(\xi^{\prime})$.

(a)(1) Pick an element $\psi\in\mathfrak{W}$. Thus there is $C^{\prime}\geq 1$ such that for every $g\in F_{N}$ we have $||\psi(g)||_{A}\leq C^{\prime}||g||_{A}$.

We first compute $[u_{n}]=\psi([w_{n}])\in S_{n}$. Thus $||u_{n}||_{A}\leq C^{\prime}||w_{n}||_{A}\leq CC^{\prime}n$, and $[u_{n}]$ is $(M,\lambda,\varepsilon)$-minimal. Therefore, by part (a) of Theorem 3.11, the Whitehead minimization algorithm on $[u_{n}]$ terminates in time $\leq K||u_{n}||_{A}\leq KCC^{\prime}n$ and and produces an element of $\mathcal{M}([u_{n}])=\mathcal{M}([w_{n}])$.

(a)(2) Let $n\geq n_{0}$. Again choose any $\psi\in\mathfrak{W}$. Then we have $w_{n}\in U_{N}(M,\lambda,\varepsilon;\psi)$. Therefore, by part (f) of Theorem 3.11, the $\psi$-speed-up of Whitehead’s algorithm decides in time at most $K\max\{|w_{n}|_{A},|u|_{A}^{2}\}\leq KC\max\{n,|u|_{A}^{2}\}$, whether or not ${\rm Aut}(F_{N})w_{n}={\rm Aut}(F_{N})u$.

(a)(3) Suppose $n\geq n_{0},m\geq m_{0}$. Thus $S_{n},S_{m}^{\prime}$ are $(M,\lambda,\varepsilon)$-minimizing and their elements are $(M,\lambda,\varepsilon)$-minimal. Choose any $\psi\in\mathfrak{W}$.

Since $\psi[w_{n}]\in S_{n}$ and $\psi[w_{m}^{\prime}]\in S_{m}^{\prime}$ and since $S_{n},S_{m}^{\prime}$ are $(M,\lambda,\varepsilon)$-minimizing, it follows that $w_{n},w_{m}^{\prime}\in U_{N}(M,\lambda,\varepsilon;\psi)$. Thus, by part (e) of Theorem 3.11, the $\psi$-speed-up of Whitehead’s algorithm decides in time at most $K\max\{|w_{n}|_{A},|w_{m}^{\prime}|_{A}\}\leq KC\max\{m,n\}$, whether or not ${\rm Aut}(F_{N})w_{n}={\rm Aut}(F_{N})w_{m}^{\prime}$.

∎

7. Group random walks as a source of $(M,\lambda,\varepsilon)$minimality

Convention 7.1* (Terminology regarding random processes).*

Let $B$ be a set with the discrete topology (such as a discrete group, the set of vertices of a graph, words in a finite alphabet, etc). For any infinite sequence of $B$-valued random variables $\mathcal{W}=W_{1},W_{2},\dots,W_{n},\dots$ we assume that the sample space $\Omega=B^{\omega}$ (as usual given the product topology for the discrete topologies on the factors $B$) is a probability space equipped with a Borel probability measure $Pr$. We will usually suppress the explicit mention of this probability measure $Pr$. Thus a trajectory of $\mathcal{W}$ is a sequence $\zeta=(w_{1},w_{2},\dots,w_{n},\dots)\in\Omega$, where all $w_{i}\in B$. We say that some property $\mathcal{E}$ holds for a.e. trajectory of $\mathcal{W}$ if

[TABLE]

Convention 7.2*.*

For a discrete probability measure $\mu:G\to[0,1]$ on a group $G$, we denote by $\langle\mbox{Supp}(\mu)\rangle_{+}$ the subsemigroup of $G$ generated by the support $\mbox{Supp}(\mu)$ of $\mu$. Note that we have $\langle\mbox{Supp}(\mu)\rangle_{+}=\cup_{n=1}^{\infty}\mbox{Supp}(\mu^{(n)})$ where $\mu^{(n)}$ is the $n$-fold convolution of $\mu$. Thus for $g\in G$ we have $g\in\langle\mbox{Supp}(\mu)\rangle_{+}$ if and only if there exist $n\geq 1$ and $g_{1},\dots,g_{n}\in G$ such that $g=g_{1}\dots g_{n}$ and $\mu(g_{i})>0$ for $i=1,\dots,n$.

Definition 7.3** (Group random walk).**

Let $G$ be a group and let $\mu:G\to[0,1]$ be a discrete probability measure on $G$. Let $X_{1},X_{2},\dots,X_{n},\dots$ be a sequence of $G$-valued i.i.d. random variables, where each $X_{i}$ has distribution $\mu$. Put $W_{n}=X_{1}\dots X_{n}\in G$, where $n=1,2\dots$. The random process

[TABLE]

is called the random walk on $G$ defined by $\mu$.

Recall that if $G$ is a group acting on a set $X$, and $\mu$ is a discrete probability measure on $G$, then a measure $\lambda$ on $X$ is called $\mu$-stationary if $\lambda=\sum_{g\in G}\mu(g)g\lambda$.

If $G$ is a non-elementary word-hyperbolic group, a discrete probability measure $\mu$ on $G$ is called non-elementary if $\langle\mbox{Supp}(\mu)\rangle_{+}$ contains some two independent loxodromic elements of $G$ (which, for a word-hyperbolic $G$ means some two elements $g_{1},g_{2}\in G$ of infinite order such that $\langle g_{1}\rangle\cap\langle g_{2}\rangle=\{1\}$).

We need the following well-known fact (see, e.g. [39, Theorem 1.1] for the most general version of this statement for random walks on groups acting on Gromov-hyperbolic spaces; see [21, Theorem 7.6] specifically for the case of a word-hyperbolic $G$):

Proposition 7.4**.**

Let $G$ be a non-elementary word-hyperbolic group and let $\mu$ be a non-elementary discrete probability measure on $G$. Let $\mathcal{W}=W_{1},W_{2},\dots,W_{n},\dots$ be the random walk on $G$ defined by $\mu$. Then:

1. (1)

For a.e. trajectory $w_{1},w_{2},\dots$ of $\mathcal{W}$ there exists $x\in\partial G$ such that $\lim_{n\to\infty}w_{n}=p$ in $G\cup\partial G$. 2. (2)

Putting, for $S\subseteq\partial G$, $\lambda(S)$ to be the probability that a trajectory of $\mathcal{W}$ converges to a point of $S$, defines a $\mu$-stationary Borel probability measure $\lambda$ on $\partial G$.

This measure $\lambda$ is called the exit measure or the hitting measure for $\mathcal{W}$.

Recall also that if $G$ is a word-hyperbolic group and $H\leq G$ is a non-elementary subgroup, then $\partial G$ contains a unique nonempty minimal closed $H$-invariant subset $\Lambda(H)\subseteq\partial G$ called the limit set of $H$ (see [30, 25] for details).

We need the following fact which seems be general folklore knowledge, although it does not seem to appear in the literature. We include a proof, explained to us by Vadim Kaimanovich, for completeness.

Proposition 7.5**.**

Let $G$ be a non-elementary word-hyperbolic group, let $\mu$ be a non-elementary discrete probability measure on $G$, and let $\lambda$ be the exit measure on $\partial G$ for the random walk on $G$ defined by $\mu$.

Suppose $H\leq G$ is a non-elementary subgroup such that $H\subseteq\langle\mbox{Supp}(\mu)\rangle_{+}$. Then $\Lambda(H)\subseteq\mbox{Supp}(\lambda)$.

In particular if $\Lambda(H)=\partial G$ then $\mbox{Supp}(\lambda)=\partial G$.

Proof.

Let $\lambda$ be the exit measure on $\partial G$ for the random walk determined by $\mu$. For any $k\geq 1$, the measure $\lambda$ is also an exit measure for the random walk based on $\mu^{(k)}$, and therefore $\lambda$ is $\mu^{(k)}$-stationary. Thus for every $n\geq 1$ we have $\lambda=\sum_{g\in G}\mu^{(n)}(g)\cdot g\lambda$. Hence $\lambda$ dominates $g\lambda$ whenever $n\geq 1$ and $\mu^{(n)}(g)>0$, that is, whenever $g\in\langle\mbox{Supp}(\mu)\rangle_{+}$. Since $H\subseteq\langle\mbox{Supp}(\mu)\rangle_{+}$, it follows that $\lambda$ dominates $h\lambda$ for every $h\in H$. Since $H$ is a subgroup of $G$, this implies that for all $h\in H$ the measures $\lambda$ and $h\lambda$ are in the same measure class. Hence for every $h\in H$ $\mbox{Supp}(\lambda)=h\mbox{Supp}(\lambda)$. Thus $\mbox{Supp}(\lambda)$ is a nonempty closed $H$-invariant subset of $\partial G$, and therefore $\Lambda(H)\subseteq\mbox{Supp}(\lambda)$, as claimed. ∎

Note that if $\langle\mbox{Supp}(\mu)\rangle_{+}$ contains a subgroup $H$ of $G$ such that $H$ has finite index in $G$, or such that $H$ is an infinite normal subgroup of $G$, then $\Lambda(H)=\partial G$ (see [30]) and therefore we get $\mbox{Supp}(\lambda)=\partial G$ in the conclusion of Proposition 7.5.

Theorem 7.6**.**

Let $F_{N}=F(A)$ be a free group of finite rank $N\geq 2$ with a free basis $A$. Let $\mu:F_{N}\to[0,1]$ be a finitely supported probability measure such that $\langle\mbox{Supp}(\mu)\rangle_{+}=F_{N}$. Let $\mathcal{W}=W_{1},W_{2},\dots$ be the random walk on $F_{N}$ defined by $\mu$. Then $\mathcal{W}$ is tame, and there exists a filling current $0\neq\nu\in\mbox{Curr}(F_{N})$ such that $\mathcal{W}$ is adapted to $\nu$.

Proof.

Let $T_{A}$ be the Cayley graph of $F_{N}$ with respect to $A$. Thus $T_{A}$ is a $2N$-regular simplicial tree. Since $\mu$ is finitely, supported, we have C:=\max\{|g|_{A}\big{|}g\in F_{N},\mu(g)>0\}<\infty. Then for all $n$ we have $|W_{n}|_{A}\leq Cn$. Thus $\mathcal{W}$ is tame.

As usual, define by $\check{\mu}:F_{N}\to[0,1]$ the probability measure on $F_{N}$ given by the formula $\check{\mu}(g)=\mu(g^{-1})$ for $g\in F_{N}$. Note that $\mbox{Supp}(\check{\mu})=(\mbox{Supp}(\mu))^{-1}=\{g^{-1}|g\in F_{N},\mu(g)>0\}$. Enumerate $\mbox{Supp}(g)$ as $\mbox{Supp}(g)=\{g_{1},\dots,g_{r}\}$ for some $r\geq 1$. Since $\langle\mbox{Supp}(\mu)\rangle_{+}=F_{N}$, for each basis element $a_{i}\in A$ (where $i=1,\dots,N$) and for each $\varepsilon\in\{\pm 1\}$ there exists a positive word $U_{i,\varepsilon}(x_{1},\dots,x_{r})$ such that $a_{i}^{\varepsilon}=U_{i,\varepsilon}(g_{1},\dots,g_{r})$ in $F_{N}$. Then $a_{i}^{-\varepsilon}=U_{i,\varepsilon}^{R}(g_{1}^{-1},\dots,g_{r}^{-1})$, where $U_{i,\varepsilon}^{R}(x_{1},\dots,x_{r})$ is the word $U_{i,\varepsilon}$ read in the reverse (but without inverting the letters). Since $g_{1}^{-1},\dots,g_{r}^{-1}\in\mbox{Supp}(\check{\mu})$, and $1\leq i\leq N$, $\varepsilon=\pm 1$ were arbitrary, it follows that $a_{1}^{\pm 1},\dots,a_{r}^{\pm 1}$ belong to $\langle\mbox{Supp}(\check{\mu})\rangle_{+}$. Hence this $\langle\mbox{Supp}(\check{\mu})\rangle_{+}=F_{N}$.

Let $\lambda$ and $\check{\lambda}$ be the exit measures on $\partial F_{N}$ for the random walks defined by $\mu$ and $\check{\mu}$ accordingly. Then, by Proposition 7.5, we have $\mbox{Supp}(\lambda)=\mbox{Supp}(\check{\lambda})=\partial F_{N}$.

The Cayley tree $T_{A}$ of $F_{N}$ is a proper $CAT(-1)$ geodesic metric space equipped with a properly discontinuous cocompact isometric action of $F_{N}$. Therefore by a result of Gekhtman [19, Theorem 1.5] there exists a geodesic current $0\neq\nu\in\mbox{Curr}(F_{N})$ such that $\mathcal{W}$ is adapted to $\nu$, and, moreover, $\nu$ belongs to the measure class $\check{\lambda}\times\lambda$ on $\partial^{2}F_{N}$. Since both $\lambda$ and $\check{\lambda}$ have full support on $\partial F_{N}$, it follows that $\nu$ has full support on $\partial^{2}F_{N}$. Therefore by [27, Corollary 1.6] the current $\nu$ is filling in $F_{N}$.

This completes the proof of Theorem 7.6.

∎

We can now conclude that algebraic and algorithmic conclusions of Theorem 6.11 and Theorem 6.12 apply in the case of $\mu$-random walks on $F_{N}$, where $\mu$ has finite support with $\langle\mbox{Supp}(\mu)\rangle_{+}=F_{N}$:

Corollary 7.7**.**

Let $F_{N}=F(A)$ be free of rank $N\geq 2$, with a free basis $A$. Let $\mu:F_{N}\to[0,1]$ be a finitely supported discrete probability measure such that $\langle\mbox{Supp}(\mu)\rangle_{+}=F_{N}$.

Let $\mathcal{W}=W_{1},\dots,W_{n},\dots$ be the random walk on $F_{N}$ defined by $\mu$.

Then there exist $M\geq 1$, $0<\varepsilon<1$ and $\lambda>1+\varepsilon$ and a subset $\mathfrak{W}\subseteq{\rm Out}(F_{N})$ with $\#\mathfrak{W}\leq M$ such that the conclusions of Theorem 6.11 and Theorem 6.12 hold for $\mathcal{W}$ with these choices of $M,\lambda,\varepsilon,\mathfrak{W}$.

8. Finite-state Markov chains and the frequency measures

We recall some basic notions and facts regarding finite state Markov chains here and refer the reader to [15, 18, 32, 33] for proofs and additional details.

8.1. Finite-state Markov chains.

Recall that a finite-state Markov chain, or FSMC $\mathcal{X}$ is defined by a finite nonempty set $S$ of states and by a family of transition probabilities $p_{\mathcal{X}}(s,s^{\prime})\geq 0$, where $s,s^{\prime}\in S$ such that for every $s\in S$ $\sum_{s^{\prime}\in S}p_{\mathcal{X}}(s,s^{\prime})=1$. Then for every integer $n\geq 1$ we also get the $n$-step transition probabilities $p_{\mathcal{X}}^{(n)}(s,s^{\prime})$ where $p_{\mathcal{X}}^{(1)}(s,s^{\prime})=p_{\mathcal{X}}(s,s^{\prime})$ and where for $n\geq 2$ and $s,s^{\prime}\in S$ we have

[TABLE]

The sample space associated with $\mathcal{X}$ is the product space $S^{\mathbb{N}}=\{\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)|s_{i}\in S\text{ for }i\geq 1\}$. The set $S$ is given the discrete topology and $S^{\mathbb{N}}$ is given the corresponding product topology, which makes $S^{\mathbb{N}}$ a compact metrizable totally disconnected topological space. For $i\geq 1$ we denote by $X_{i}:S^{\mathbb{N}}\to S$ the function picking out the $i$-th coordinate of an element of $S^{\mathbb{N}}$. The transition matrix $M=M(\mathcal{X})$ is an $S\times S$ matrix where for $s,s^{\prime}\in S$ the entry $M(s,s^{\prime})$ of $M$ is defined as $M(s,s^{\prime})=p_{\mathcal{X}}^{(n)}(s,s^{\prime})$. Thus $M(\mathcal{X})$ is a nonnegative matrix, where the sum of the entries in each row is equal to $1$. Also, for all $n\geq 1$ and $s,s^{\prime}\in S$ we have $p_{\mathcal{X}}^{(n)}(s,s^{\prime})=(M^{n})(s,s^{\prime})$. A FSMC $\mathcal{X}$ as above is called irreducible if for all $s,s^{\prime}\in S$ there exists $n\geq 1$ such that $p_{\mathcal{X}}^{(n)}(s,s^{\prime})>0$. Thus $\mathcal{X}$ is irreducible if and only if the nonnegative matrix $M(\mathcal{X})$ is irreducible in the sense of Perron-Frobenius theory.

For an FSMC $\mathcal{X}$, given a initial probability distribution $\mu$ on $S$, we obtain the corresponding Markov Process $\mathcal{X}_{\mu}=X_{1},\dots,X_{n},\dots$ where each $X_{i}$ is an $S$-valued random variable with probability distribution $\mu_{i}$ on $S$, where $\mu_{1}=\mu$ and where for $i\geq 2$ and $s^{\prime}\in S$ we have $\mu_{i}(s^{\prime})=\sum_{s\in S}\mu_{i-1}(s)p_{\mathcal{X}}(s,s^{\prime})$. An initial distribution $\mu$ on $S$ is called stationary for $\mathcal{X}$ if $\mu_{i}=\mu$ for all $i\geq 1$ (equivalently, if $\mu_{2}=\mu$). It is well-known, by the basic result of Perron-Frobenius theory, that if $\mathcal{X}$ is an irreducible finite-state Markov chain with state set $S$, then there is a unique stationary probability distribution $\mu$ on $S$ for $\mathcal{X}$, and that it satisfies $\mu(s)>0$ for all $s\in S$. In this case the vector $(\mu(s))_{s\in S}$ is the Perron-Frobenius eigenvector of $||.||_{1}$-norm $1$ for the matrix $M(\mathcal{X})$ with eigenvalue $\lambda=1$, and, moreover, $\lambda=1$ is the Perron-Frobenius eigenvalue for $M(\mathcal{X})$. In particular, the eigenvalue $\lambda=1$ is simple and is equal to the spectral radius of $M(\mathcal{X})$.

For an FSMC $\mathcal{X}$ with state set $S$, a word $w=s_{1}\dots s_{n}\in S^{n}$ of length $n\geq 2$ is called feasible if $p_{\mathcal{X}}(s_{1},s_{2})\dots p_{\mathcal{X}}(s_{n-1},s_{n})>0$. Also, we consider all words $w=s\in S$ of length $n=1$ to be feasible. (Hence every nonempty subword of a feasible word is also feasible). An element $\xi=(s_{1},s_{2},\dots)\in S^{\mathbb{N}}$ is feasible for $\mathcal{X}$ if for every $n\geq 1$ the word $s_{1}\dots s_{n}$ is feasible. Denote by $(S^{\mathbb{N}})_{+}$ the set of all feasible $\xi\in S^{\mathbb{N}}$. Also, for every $n\geq 1$ denote by $(S^{n})_{+}$ the set of all feasible $s_{1}\dots s_{n}\in S^{n}$.

For a word $w=s_{1}\dots s_{n}\in S^{n}$ (where $n\geq 2$) put

[TABLE]

Any initial probability distribution $\mu$ on $S$ defines a Borel probability $\mu_{\infty}$ via the standard convolution formulas. Namely, if $n\geq 1,s_{1},\dots s_{n}\in S$ then

[TABLE]

where $Cyl(s_{1}\dots s_{n})=\{\xi\in S^{\mathbb{N}}|X_{i}(\xi)=s_{i}\text{ for }i=1,\dots,n\}$.

If $\mu$ is strictly positive on $S$, then the support $supp(\mu_{\infty})$ of $\mu_{\infty}$ is equal to $(S^{\mathbb{N}})_{+}$. In particular, that is the case if $\mathcal{X}$ is an irreducible FSMC and $\mu$ is the unique stationary probability distribution on $S$.

Definition 8.1** (Occurrences and frequencies).**

Let $\mathcal{X}$ be an irreducible finite-state Markov chain with state set $S$.

(1) For a word $w=s_{1}\dots s_{n}\in S^{n}$ (where $n\geq 1$) and an element $s\in S$ we denote by $\langle s,w\rangle$ the number of those $i\in\{1,\dots,n\}$ such that $s_{i}=s$. We call $\langle s,w\rangle$ the number of occurrences of $s$ in $w$. We also put $\theta_{s}(w)=\frac{\langle s,w\rangle}{|w|}$, where $|w|=n$ is the length of $w$. We call $\theta_{s}(w)$ the frequency of $s$ in $w$.

(2) The above notions can be extended from $s$ to arbitrary nonempty words $v\in S^{\ast}$ as follows. Let $v=y_{1}\dots y_{m}\in S^{m}$ where $y_{j}\in S$ for $j=1,\dots,m$. Also denote by $w^{\infty}$ the semi-infinite word $w^{\infty}=wwww\dots$. For an arbitrary integer $i\geq 1$ we still denote by $s_{i}\in S$ the $i$-th letter of $w^{\infty}$. Now define $\langle v,w\rangle$ to be the number of $i\in\{1,\dots,n\}$ such that in $w^{\infty}$ we have $s_{i}=y_{1},s_{i+1}=y_{2},\dots,s_{i+m-1}=y_{m}$. We call $\langle v,w\rangle$ the number of occurrences of $v$ in $w$, and we call $\theta_{v}(w)=\frac{\langle v,w\rangle}{|w|}$ the frequency of $v$ in $w$.

We record the following immediate corollary of the above definition (which holds since we defined the numbers of occurrences in $w$ cyclically).

Lemma 8.2**.**

Let $w\in S^{n}$ where $n\geq 1$. Then the following hold:

1. (1)

We have $n=|w|=\sum_{s\in S}\langle s,w\rangle$ and $1=\sum_{s\in S}\theta_{s}(w)$. 2. (2)

For every $m\geq 1$ we have $n=|w|=\sum_{v\in S^{m}}\langle v,w\rangle$ and $1=\sum_{v\in S^{m}}\theta_{v}(w)$. 3. (3)

For every $m\geq 1$ and every $v\in S^{m}$ we have

[TABLE]

and

[TABLE]

For a finite-state Markov chain $\mathcal{X}$ with state set $S$ and an element $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)$ of $S^{\mathbb{N}}$, we denote $w_{n}=s_{1}\dots s_{n}\in S^{n}$, where $n\geq 1$.

The Strong Law of Large numbers applies to a finite-state Markov chain implies:

Proposition 8.3**.**

Let $\mathcal{X}$ be an irreducible finite-state Markov chain with state set $S$ and let $\mu_{0}$ be the unique stationary probability distribution on $S$. Let $\mu$ be an arbitrary initial distribution on $S$ defining the corresponding Markov process $\mathcal{X}_{\mu}=X_{1},\dots,X_{n},\dots$. Then the following hold:

1. (1)

For every $s\in S$ and for $\mu_{\infty}$-a. e. trajectory $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)\in S^{\mathbb{N}}$ of $\mathcal{X}_{\mu}$, we have

[TABLE] 2. (2)

For every $0<\varepsilon\leq 1$ and every $s\in S$

[TABLE]

and the convergence in this limit is exponentially fast as $n\to\infty$.

8.2. Iterated Markov Chains

Let $\mathcal{X}$ be a finite-state Markov chain with state set $S$. Let $k\geq 1$ be an integer. Consider a finite-state Markov chain $\mathcal{X}[k]$ with the state set $(S^{k})_{+}$ and with transition probabilities defined as follows. Suppose $s_{1}\dots s_{k}\in(S^{k})_{+}$ and $s\in S$ are such that $p_{\mathcal{X}}(s_{k},s)>0$ (so that $s_{1}\dots s_{k}s\in S^{k+1}$ is feasible for $\mathcal{X}$, and $s_{2}\dots s_{k}s\in(S^{k})_{+}$). Then put $p_{\mathcal{X}[k]}(s_{1}\dots s_{k},s_{2}\dots s_{k}s)=p_{\mathcal{X}}(s_{k},s)$. Set all other transition probabilities in $\mathcal{X}[k]$ to be [math]. Note that we have $\mathcal{X}[1]=\mathcal{X}$.

It is not hard to see that if $\mathcal{X}$ as above is irreducible then for every $k\geq 1$ the FSMC $\mathcal{X}[k]$ is also irreducible. Moreover, in this case there is a natural canonical homeomorphism between the set of infinite feasible trajectories $(S^{N})_{+}$ of $\mathcal{X}$ and the set $\left(((S^{k})_{+})^{\mathbb{N}}\right)_{+}$ of infinite feasible trajectories for $\mathcal{X}[k]$. Under this homeomorphism a sequence $\xi=(s_{1},\dots,s_{n}\dots)\in(S^{\mathbb{N}})_{+}$ goes to $(v_{1},v_{2}\dots,v_{n},\dots)\in\left(((S^{k})_{+})^{\mathbb{N}}\right)_{+}$ where $v_{i}=s_{i}s_{i+1}\dots s_{i+k-1}$. Moreover, if $\mu_{0}$ is the unique stationary distribution for $\mathcal{X}$ on $S$ then

[TABLE]

where $s_{1}\dots s_{k}\in(S^{k})_{+}$, is the unique stationary probability distribution for $\mathcal{X}[k]$. Using these facts and the application of Proposition 8.3, standard results about Markov chains imply the following statement; see [6, Proposition 3.13] for a more detailed version of this statement, with explicit speed of convergence estimates:

Proposition 8.4**.**

Let $\mathcal{X}$ be a finite-state Markov chain with state set $S$ and let $\mu_{0}$ be the unique stationary probability distribution on $S$. Let $\mu$ be an arbitrary strictly positive initial distribution on $S$ defining the corresponding Markov process $\mathcal{X}_{\mu}=X_{1},\dots,X_{n},\dots$. Let $k\geq 1$ be an integer and let $\mu_{0}[k]$ be the distribution on $(S^{k})_{+}$ defined by () above. We extend $\mu_{0}[k]$ to $S^{k}$ by setting $\mu_{0}[k](v)=0$ for all $v\in S^{k}\setminus(S^{k})_{+}$.*

Then the following hold:

1. (1)

For every $v\in S^{k}$ and for $\mu_{\infty}$-a. e. trajectory $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)\in S^{\mathbb{N}}$ of $\mathcal{X}_{\mu}$, we have

[TABLE] 2. (2)

For every $0<\varepsilon\leq 1$ and every $v\in S^{k}$

[TABLE]

and the convergence in this limit is exponentially fast as $n\to\infty$.

∎

Corollary 8.5**.**

Let $X$, $S$, $\mu_{0}$ and $\mu$ be as in Proposition 8.4 above. Then:

1. (1)

For every $m\geq 1$ we have $1=\sum_{v\in S^{m}}\mu_{0}[k](v)$. 2. (2)

For every $m\geq 1$ and every $v\in S^{m}$ we have

[TABLE]

Proof.

Take a random trajectory $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)\in S^{\mathbb{N}}$ of $\mathcal{X}_{\mu}$ to which the conclusion of Proposition 8.4 applies and put $w_{n}=s_{1}\dots s_{n}$ for all $n\geq 1$. The conclusion of part (1) of the corollary now follows directly from part (1) of Proposition 8.4 and from part (1) of Lemma 8.2.

Now let $m\geq 1$ and let $v\in S^{m}$. Then by part (3) of Lemma 8.2 we have

[TABLE]

By passing to the limit as $n\to\infty$ and applying part (1) of Proposition 8.4 , we obtain part (2) of the corollary. ∎

8.3. Quasi-inversions

We also need the following, somewhat technical to state but mathematically fairly straightforward, statement to later rule out the situation where a random reduced path in a finite graph is closed but far from being cyclically reduced.

We say that a FSMC $\mathcal{X}$ with state set $S$ is tight if $p_{\mathcal{X}}(s,s^{\prime})<1$ for all $s,s^{\prime}\in S$.

Let $\mathcal{X}$ be a FSMC with state set $S$ where $\#S\geq 2$. Let $\iota:S^{\prime}\to S$ be an injective function where $S^{\prime}\subseteq S$. For a word $w\in(S^{\prime})^{\ast}$, $w=s_{1}\dots s_{n}$ with $s_{i}\in S^{\prime}$ put $\iota(w)=\iota(s_{1})\dots\iota(s_{n})$. For $w\in S^{\ast}\setminus(S^{\prime})^{\ast}$ put $\iota(w)=\varepsilon$, the empty word. Also, for a word $w\in S^{\ast}$ denote by $w^{R}$ the reverse word. That is, if $w=s_{1}\dots s_{n}$ with $s_{i}\in S$ then $w^{R}=s_{n}\dots s_{1}$.

Proposition 8.6**.**

Let $\mathcal{X}$ be an irreducible tight finite-state Markov chain with state set $S$ where $\#S\geq 2$, and let $\mu_{0}$ be the unique stationary probability distribution on $S$. Put $\sigma=\max_{s,s^{\prime}}p_{\mathcal{X}}(s,s^{\prime})$ (so that $0<\sigma<1$).

Let $\mu$ be an arbitrary initial distribution on $S$ defining the corresponding Markov process $\mathcal{X}_{\mu}=X_{1},\dots,X_{n},\dots$. Let $\iota:S^{\prime}\to S$ be an injective function where $S^{\prime}\subseteq S$ (with $\iota:S^{\ast}\to S^{\ast}$ extended as above as well). Then the following hold for a trajectory $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)\in S^{\mathbb{N}}$ of $\mathcal{X}_{\mu}$:

1. (1)

We have

[TABLE]

and, in particular,

[TABLE] 2. (2)

Let $a,b\in S$ be two states. Then for the conditional probability, conditioned on $s_{1}=a,s_{n}=b$, we have:

[TABLE]

and, in particular,

[TABLE]

Proof.

(1) For a trajectory $\xi=(s_{1},s_{2},s_{3},\dots,s_{n},\dots)\in S^{\mathbb{N}}$ of $\mathcal{X}_{\mu}$ denote $w_{n}=w_{n}(\xi)=s_{1}\dots s_{n}$. For $m\leq n$ denote by $\alpha_{m}(w_{n})$ the initial segment of $w_{n}$ of length $m$.

Let $n\geq 1$ and let $E_{n}$ be the event that $s_{1}s_{2}\dots s_{\lfloor\sqrt{n}\rfloor}=\left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_{n})\right)^{R}$. For $u=y_{1}\dots y_{n-\lfloor\sqrt{n}\rfloor}\in S^{n-\lfloor\sqrt{n}\rfloor}$ let $t(u)=y_{n-\lfloor\sqrt{n}\rfloor}\in S$ be the last letter of $u$.

For any fixed $u=y_{1}\dots y_{n-\lfloor\sqrt{n}\rfloor}\in S^{n-\lfloor\sqrt{n}\rfloor}$ the conditional probability $Pr(w_{n}\in E_{n}|w_{n-\lfloor\sqrt{n}\rfloor}=u)$ is equal to

[TABLE]

Then

[TABLE]

Thus part (1) is verified. The proof of part (2) is similar and we leave the details to the reader.

∎

Remark 8.7*.*

In fact, the assumption that $\mathcal{X}$ be tight is not crucial in Proposition 8.6 and a similar result holds if we assume that $X$ is an irreducible FSMC with $\#S\geq 2$. We make the tightness assumption to simplify the argument.

9. Graph-based non-backtracking random walks

Convention 9.1*.*

In this section we will assume that $F_{N}=F(A)$ is a free group of finite rank $N\geq 2$, that $\Gamma$ is a finite connected oriented graph with all vertices of degree $\geq 3$ and with the first betti number $b(\Gamma)=N$, and that $\alpha:F_{N}\to\pi_{1}(\Gamma,x_{0})$ is a fixed isomorphism, where $x_{0}\in V\Gamma$ is some base-vertex. We equip $\Gamma$ and $T_{0}=\widetilde{\Gamma}$ with simplicial metrics, where every edge has length $1$.

Note that for $\Gamma$ as above we always have $\#E\Gamma\leq 6N$.

Definition 9.2**.**

Under the above convention, a FSMC $\mathcal{X}$ with state set $S$ is $\Gamma$-based if the following hold:

1. (1)

We have $S\subseteq E\Gamma$, with $\#S\geq 2$. 2. (2)

Whenever $e,e^{\prime}\in S$ are such that $p_{\mathcal{X}}(e,e^{\prime})>0$ then $t(e)=o(e^{\prime})$ in $\Gamma$ and $e^{\prime}\neq e^{-1}$.

Thus for a $\Gamma$-based FSMC $\mathcal{X}$ as above, the space of feasible trajectories $(S^{\mathbb{N}})_{+}$ can be thought of as a subset of the set $\Omega(\Gamma)$ of all reduced semi-infinite edge-paths $\gamma=e_{1},e_{2},\dots$ in $\Gamma$. Similarly, $(S^{n})_{+}$ can be thought of as a subset of the set $\Omega_{n}(\Gamma)$ of all reduced length $n$ edge-paths $e_{1},e_{2},\dots,e_{n}$ in $\Gamma$.

Proposition 9.3**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu_{0}$ be the unique stationary probability distribution on $S$. For every $k\geq 1$ we extend $\mu_{0}[k]$ to $\Omega_{k}(\Gamma)$ by setting $\mu_{0}[k](v)=0$ for every $v\in\Omega_{k}(\Gamma)-(S^{k})_{+}$.

There exists a unique geodesic current $\nu$ on $F_{N}$ with the following properties:

1. (1)

For every $k\geq 1$ and every $v\in\Omega_{k}(\Gamma)$ we have $\langle v,\nu\rangle_{\Gamma}=\mu_{0}[k](v)+\mu_{0}[k](v^{-1})$. 2. (2)

We have $\langle T_{0},\nu\rangle=1$.

Proof.

We use the formulas in part (1) of the proposition to define a system of weights $\nu$ on $\cup_{n\geq 1}\Omega_{n}(\Gamma)$. Note that these weights are already symmetrized since the defining equations for the weights in (1) give the same answers for $v$ and $v^{-1}$. Now Corollary 8.5 implies that these $\nu$ weights satisfy the switch conditions. Therefore they do define a geodesic current $\nu\in\mbox{Curr}(F_{N})$.

Also, part (1) of Corollary 8.5 implies that $\sum_{e\in E\Gamma}\mu_{0}[1](e)=1=\sum_{e\in E\Gamma}\mu_{0}[1](e^{-1})$ and therefore

[TABLE]

Thus part (1) of the proposition holds and, in particular $\nu\neq 0$ in $\mbox{Curr}(F_{N})$. ∎

Definition 9.4** (Characteristic current).**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $0\neq\nu\in\mbox{Curr}(F_{N})$ be the geodesic current constructed in Proposition 9.3 above. We call $\nu$ the characteristic current of $\mathcal{X}$ and denote it $\nu=\nu_{\mathcal{X}}$

Definition 9.5** ($\mathcal{X}$-directed random walk on $\Gamma$).**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$ defining the corresponding Markov process $\mathcal{X}_{\mu}=X_{1},\dots,X_{n},\dots$. For every $n=1,2,\dots$ put $W_{n}=X_{1}\dots X_{n}$ so that $W_{n}$ takes values in $S^{n}$. The random process $\mathcal{W}_{\mu}=W_{1},\dots,W_{n},\dots$ is called the $\mathcal{X}$-directed non-backtracking random walk on $\Gamma$ corresponding to $\mu$.

Note that for $\mathcal{W}_{\mu}$ and any $n\geq 1$ the only feasible values of $X_{n}$ are contained in $S^{n}\cap\Omega_{n}(\Gamma)$.

Since in general $\Gamma$ may have more than one vertex, a reduced edge-path in $\Gamma$ (such as, for example, the length-$n$ path given by $W_{n}$ in the above setting) is not necessarily closed and thus may not define a conjugacy class in $\pi_{1}(\Gamma,x_{0})$. To get around this issue, we modify $\mathcal{W}_{\mu}$ slightly, in two different ways to output closed paths in $\Gamma$.

Definition 9.6** (Closing path system).**

Let $\Gamma$ be as in Convention 9.1. A closing path system for $\Gamma$ is a family $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ of reduced edge-paths in $\Gamma$ such that for every $e,e^{\prime}\in E\Gamma$ $e\beta_{e,e^{\prime}}e^{\prime}$ is a reduced edge-path in $\Gamma$.

For a non-degenerate reduced edge-path $\gamma$ in $\Gamma$ define the $\mathcal{B}$-closing $\widehat{\gamma}$ of $\gamma$ as $\widehat{\gamma}=\gamma\beta{e,e^{\prime}}$ where $e$ is the last edge of $\gamma$ and $e^{\prime}$ is the first edge of $\gamma$. Note also that for any nondegenerate reduced edge-path $\gamma$ in $\Gamma$ the $\mathcal{B}$-closing $\widehat{\gamma}$ is a reduced and cyclically reduced closed edge-path in $\Gamma$.

Note that $\mathcal{B}$ is above, if $e,e^{\prime}\in E\Gamma$ then $t(e)=o(\beta_{e,e^{\prime}})$ and $o(e^{\prime})=t(\beta_{e,e^{\prime}})$. It is also easy to see that for every $\Gamma$ some closing path system $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ exists, and we can always choose $\mathcal{B}$ so that $|\beta_{e,e^{\prime}}|\leq|E\Gamma|\leq 6N$ for all $e,e^{\prime}\in E\Gamma$.

Definition 9.7** ($\mathcal{B}$-closing of a non-backtracking walk on $\Gamma$).**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\mu$ be any initial probability distribution on $S$ and let $\mathcal{W}_{\mu}=W_{1},\dots,W_{n},\dots$ be the $\mathcal{X}$-directed non-backtracking random walk on $\Gamma$ corresponding to $\mu$. Define the random process $\widehat{\mathcal{W}_{\mu}}=\widehat{W}_{1},\dots,\widehat{W}_{n},\dots$, where $\widehat{W}_{n}$ the $\mathcal{B}$-closing of $W_{n}$. We call $\widehat{\mathcal{W}_{\mu}}$ the $\mathcal{B}$-closing of $\mathcal{W}_{\mu}$.

An advantage of using $\widehat{\mathcal{W}_{\mu}}$ is that it always outputs reduced and cyclically reduced closed paths $\widehat{W}_{n}$ of length $n\leq|\widehat{W}_{n}|\leq n+C$, where $C=\max_{e,e^{\prime}}|\beta_{e,e^{\prime}}|$. However, a weakness of this approach is that the path $W_{n}$ is already a closed edge-path in $\Gamma$, with asymptotically positive probability as $n\to\infty$ (if $\mathcal{X}$ is irreducible and $\Gamma$-based). Therefore we offer a variation of the $\widehat{\mathcal{W}_{\mu}}$ approach which takes this fact into account.

For a reduced nondegenerate closed edge-path $\gamma$ in $\Gamma$ denote by $cyc(\gamma)$ the subpath of $\gamma$ obtained from $\gamma$ by a maximal cyclic reduction. Thus $cyc(\gamma)$ is a nondegenerate closed reduced and cyclically reduced edge-path in $\Gamma$.

Notation 9.8*.*

Let $\mathcal{B}$ be a closing path system for $\Gamma$. For a nondegenerate reduced edge-path $\gamma$ in $\Gamma$ let $\breve{\gamma}:=cyc(\gamma)$ is $\gamma$ is a closed path, and let $\breve{\gamma}:=\widehat{\gamma}$ otherwise. Thus in both cases $\breve{\gamma}$ is a closed reduced and cyclically reduced edge-path in $\Gamma$ (but it may now have length $<n$). We call $\breve{\gamma}$ the modified $\mathcal{B}$-closing of $\gamma$.

Definition 9.9**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal{W}_{\mu}=W_{1},\dots,W_{n},\dots$ be the $\mathcal{X}$-directed non-backtracking random walk on $\Gamma$ corresponding to $\mu$.

Define the random process $\breve{\mathcal{W}_{\mu}}=\breve{W}_{1},\dots,\breve{W}_{n},\dots$, where $\breve{W}_{n}$ the modified $\mathcal{B}$-closing of $W_{n}$. We call $\breve{\mathcal{W}_{\mu}}$ the modified $\mathcal{B}$-closing of $\mathcal{W}_{\mu}$.

Remark 9.10*.*

It is easy to see that the random processes $\widehat{\mathcal{W}_{\mu}}$, $\breve{\mathcal{W}_{\mu}}$ considered above always satisfy condition (1) of Definition 6.9.

Theorem 9.11**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_{\mathcal{X}}\in\mbox{Curr}(F_{N})$ be the characteristic current for $\mathcal{X}$. Let $\widehat{\mathcal{W}_{\mu}}=\widehat{W}_{1},\dots,\widehat{W}_{n},\dots$ be the $\mathcal{B}$-closing of $\mathcal{W}_{\mu}$.

Then $\widehat{\mathcal{W}_{\mu}}$ is tame and adapted to $\nu_{\mathcal{X}}$.

Proof.

Put $C=\max_{e,e^{\prime}}|\beta_{e,e^{\prime}}|$. We have $|\widehat{W}_{n}|\leq n+C$ for all $n\geq 1$, which implies that $\widehat{\mathcal{W}_{\mu}}$ is tame.

Let $\xi=e_{1},\dots,e_{n},\dots$ be a random trajectory of $\mathcal{X}_{\mu}$, where $e_{i}\in S\subseteq E\Gamma$ for all $i\geq 1$. Denote $w_{n}=e_{1}\dots e_{n}$, so that the $\mathcal{B}$-closing of $w_{n}$ is $\widehat{w}_{n}=e_{1}\dots e_{n}\beta_{e_{n},e_{1}}$. Thus $\widehat{w}_{n}$ is a closed reduced and cyclically reduced edge-path in $\Gamma$ with $n\leq|\widehat{w}_{n}|\leq n+C$. We can then also think, via the marking isomorphism, of $\widehat{w}_{n}$ as defining a nontrivial conjugacy class in $F_{N}$. Recall that for a nontrivial reduced edge-path $v$ in $\Gamma$ of length $|v|=k\geq 1$ we have $\langle v,\eta_{\widehat{w}_{n}}\rangle_{\Gamma}=\langle v,\widehat{w}_{n}\rangle+\langle v^{-1},\widehat{w}_{n}\rangle$, where the latter two terms are the numbers of occurrences of $v^{\pm 1}$ in $\widehat{w}_{n}$ in the sense of Definition 8.1. Now Proposition 8.4 implies that

[TABLE]

Therefore $\lim_{n\to\infty}\frac{1}{n}\eta_{\widehat{w}_{n}}=\nu_{\mathcal{X}}$ in $\mbox{Curr}(F_{N})$, and hence $\lim_{n\to\infty}[\eta_{\widehat{w}_{n}}]=[\nu_{\mathcal{X}}]$ in $\mathbb{P}\mbox{Curr}(F_{N})$. This means that $\widehat{\mathcal{W}_{\mu}}$ is adapted to $\nu_{\mathcal{X}}$, as required. ∎

Theorem 9.12**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_{\mathcal{X}}\in\mbox{Curr}(F_{N})$ be the characteristic current for $\mathcal{X}$. Let $\breve{\mathcal{W}_{\mu}}=\breve{W}_{1},\dots,\breve{W}_{n},\dots$ be the modified $\mathcal{B}$-closing of $\mathcal{W}_{\mu}$.

Then $\breve{\mathcal{W}_{\mu}}$ is tame and adapted to $\nu_{\mathcal{X}}$.

Proof.

Again put $C=\max_{e,e^{\prime}}|\beta_{e,e^{\prime}}|$. We have $|\breve{W}_{n}|\leq n+C$ for all $n\geq 1$, which implies that $\breve{\mathcal{W}_{\mu}}$ is tame.

Let $\xi=e_{1},\dots,e_{n},\dots$ be a random trajectory of $\mathcal{X}_{\mu}$, where $e_{i}\in S\subseteq E\Gamma$ for all $i\geq 1$. Denote $w_{n}=e_{1}\dots e_{n}$. Thus the corresponding trajectory of $\breve{\mathcal{W}_{\mu}}$ is $\breve{w}_{1},\breve{w}_{2},\dots,\breve{w}_{n},\dots$. We need to show that $\lim_{n\to\infty}[\eta_{\breve{w}_{n}}]=[\nu_{\mathcal{X}}]$ in $\mathbb{P}\mbox{Curr}(F_{N})$. For every $n\geq 1$ such that $w_{n}$ is a non-closed path, we have $\breve{w}_{n}=\widehat{w}_{n}$, and the conclusion of Proposition LABEL:p:cl applies. Thus it remains to show that for any infinite increasing sequence $n_{i}$ of indices such that $w_{n_{i}}$ is a closed path we have $\lim_{i\to\infty}[\eta_{\breve{w}_{n_{i}}}]=[\nu_{\mathcal{X}}]$ in $\mathbb{P}\mbox{Curr}(F_{N})$.

Let $n_{i}$ be such a sequence. Then for all $i\geq 1$ we have $\breve{w}_{n_{i}}=cyc(w_{n_{i}})$.

Let $S_{1}=\{e\in S|\overline{e}\in S\}$ and define $\iota:S_{1}\to S$ as $\iota(e)=e^{-1}$ for $e\in S_{1}$. Since $\mathcal{X}$ is tight, Proposition 8.6 implies that we have

[TABLE]

for $i\to\infty$. Recall also that $cyc(w_{n_{i}})$ is a subpath of $w_{n_{i}}$.

Let $v$ be an arbitrary nondegenerate reduced edge-path in $\Gamma$ of length $k\geq 1$. Then we have

[TABLE]

and

[TABLE]

Then

Now Proposition 8.4 implies that

[TABLE]

Therefore $\lim_{i\to\infty}\frac{1}{n_{i}}\eta_{\widehat{w}_{n_{i}}}=\nu_{\mathcal{X}}$ in $\mbox{Curr}(F_{N})$, and hence $\lim_{i\to\infty}[\eta_{\widehat{w}_{n_{i}}}]=[\nu_{\mathcal{X}}]$ in $\mathbb{P}\mbox{Curr}(F_{N})$. As noted above, this implies that $\breve{\mathcal{W}_{\mu}}$ is adapted to $\nu_{\mathcal{X}}$, as required.

∎

In summary, we get:

Corollary 9.13**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal{B}=(\beta_{e,e^{\prime}})_{e,e^{\prime}\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_{\mathcal{X}}\in\mbox{Curr}(F_{N})$ be the characteristic current for $\mathcal{X}$. Suppose that $\nu_{X}$ is filling in $F_{N}$.

Then $\widehat{W}_{\mu}$ and $\breve{W}_{\mu}$ are adapted to the characteristic current $\nu_{X}$. Therefore Theorem 6.11 and Theorem 6.12 apply to $\widehat{W}_{\mu}$ and $\breve{W}_{\mu}$.

We next explain several situations where one can guarantee that the current $\nu_{\mathcal{X}}\in\mbox{Curr}(F_{N})$ is filling.

Proposition 9.14**.**

Let $\mathcal{X}$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\nu_{\mathcal{X}}\in\mbox{Curr}(F_{N})$ be the characteristic current for $\mathcal{X}$.

1. (1)

Suppose that $\mathcal{X}$ has the property that $S=E\Gamma$ and that for every $e,e^{\prime}\in E\Gamma$ such that $ee^{\prime}$ is a reduced edge-path in $\Gamma$ we have $p_{\mathcal{X}}(e,e^{\prime})>0$. Then the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling. 2. (2)

Suppose that $\Gamma=R_{A}$, the $N$-rose corresponding to a free basis $A=\{a_{1},\dots,a_{N}\}$ of $F_{N}$ (so that we can identify $E(R_{A})=A^{\pm 1}$). Suppose that $\mathcal{X}$ is such $A\subseteq S$ and that for all $1\leq i,j\leq N$ we have $p_{\mathcal{X}}(a_{i},a_{j})>0$. Then the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling. 3. (3)

Suppose there exists a nondegenerate reduced cyclically reduced closed edge-path $w$ in $\Gamma$ such that $w$ represents a filling element of $F_{N}$ and that for every $n\geq 2$ we have $p_{\mathcal{X}}(w^{n})>0$. Then the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling. 4. (4)

Suppose there exists a free basis $A=\{a_{1},\dots,a_{k}\}$ such that the following hold. For $i=1,\dots,N$ let $w_{i}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_{i}$ in $F_{N}$. For $1\leq i<j\leq N$ let $w_{i,j}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_{i}$ in $F_{N}$. Suppose that we have $p_{\mathcal{X}}(w_{i}^{2})>0$ for $i=1,\dots,N$ and that we have $p_{\mathcal{X}}(w_{ij}^{2})>0$ for all $1\leq i<j\leq N$. Then the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling.

Proof.

Let $\mu_{0}$ be the unique stationary probability distribution on $S$ for $\mathcal{X}$.

(1) The assumption on $\mathcal{X}$ implies that for every reduced edge-path $v$ in $\Gamma$ of length $k\geq 1$ we have $\mu_{0}[k](v)>0$, and therefore, by definition of $\nu_{\mathcal{X}}$, we also have $\langle v,\nu_{\mathcal{X}}\rangle_{\Gamma}>0$. Thus $\nu_{X}\in\mbox{Curr}(F_{N})$ has full support and therefore $\nu_{X}$ is filling in $F_{N}$.

(2) The assumptions on $\mathcal{X}$ (with $\Gamma=R_{A}$) imply that for all $n\geq 1$ and all $1\leq i,j\leq N$ we have $\mu_{0}[n](a_{i}^{n}),\mu_{0}[n](a_{j}^{n}),\mu_{0}[2n]((a_{i}a_{j})^{n})>0$. Therefore, by definition of $\nu_{X}$, we also have $\langle a^{i},\nu_{\mathcal{X}}\rangle_{A}>0$ and $\langle(a_{i}a_{j})^{n},\nu_{\mathcal{X}}\rangle_{A}>0$. Therefore, by Proposition 5.7, the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling.

(3) Again, similarly to (1) and (2) we see that for every $n\geq 1$ $\langle w^{n},\nu_{\mathcal{X}}\rangle_{\Gamma}>0$. Therefore by Corollary 5.6 the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling.

(4) Recall that for a reduced edge-path $v$ in $\Gamma$ of length $k\geq 2$ and starting with $e_{1}\in E\Gamma$ we have $\mu_{0}[k](v)=\mu_{0}(e_{1})p_{\mathcal{X}}(v)$. Thus $\mu_{0}[k](v)>0$ if and only if $e_{1}\in S$ and the transition probabilities $p_{\mathcal{X}}(e^{\prime},e^{\prime\prime})$ are $>0$ for all length-2 subpaths $e^{\prime}e^{\prime\prime}$ of $v$. Note also that if for the 2-nd edge $e_{2}$ of $v$ we have $p_{\mathcal{X}}(e_{1},e_{2})>0$ then $e_{1},e_{2}\in S$.

Hence the assumptions in part (4) imply that for $i=1,\dots,N$ and all $n\geq 1$ we have $\mu_{0}[n](w_{i}^{n})>0$, and, similarly, for all $1\leq i<j\leq N$ and all $n\geq 1$ we have $\mu_{0}[n](w_{i,j}^{n})>0$. Therefore, by definition of $\nu_{X}$, we have $\langle w_{i}^{n},\nu_{\mathcal{X}}\rangle_{\Gamma}>0$, and, also, for all $1\leq i<j\leq N$ and all $n\geq 1$ we have $\langle w_{ij}^{n},\nu_{\mathcal{X}}\rangle_{\Gamma}>0$. Therefore, by Proposition 5.7, the current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling.

∎

Example 9.15*.*

Let $A=\{a_{1},\dots,a_{N}\}$ be a free basis of $F_{N}=F(A)$ and let $\Gamma=R_{A}$ be the corresponding $N$-rose.

(1) Consider an $R_{A}$-based FSMC $\mathcal{X}$ with state set $S=A^{\pm 1}$ and transition probabilities $p_{\mathcal{X}}(a_{i}^{\varepsilon},a_{j}^{\delta})=\frac{1}{2N-1}$ if $a_{i}^{\varepsilon}\neq a_{j}^{-\delta}$ and $p_{\mathcal{X}}(a_{i}^{\varepsilon},a_{i}^{-\varepsilon})=0$, where $\varepsilon,\delta=\pm 1$. Then $\mathcal{X}$ is irreducible and tight. The stationary distribution $\mu_{0}$ is the uniform probability distribution on $A^{\pm 1}$. Then $\mathcal{X}$, with an initial distrubution $\mu$ on $A^{\pm 1}$, defines the standard non-backtracking simple random walk $\mathcal{W}_{\mu}=W_{1},W_{2}\dots$ on $F_{N}=F(A)$. In this case the characteristic current $\nu_{\mathcal{X}}$ is the uniform current $\nu_{A}$ corresponding to $A$. The current $\nu_{\mathcal{X}}=\nu_{A}$ has full support and therefore is filling. Since $R_{A}$ has only one vertex, the edge-path $W_{n}$ is always closed, and we have $\breve{W}_{n}=W_{n}$. Using a closing path system $\mathcal{B}$ produces cyclically reduced words $\widehat{W}_{n}=W_{n}\beta$, where $\beta\in\mathcal{B}$ is an appropriate closing path. The fact that $W_{n}$ is adapted to $\nu_{A}$ is explained in more detail in [24] and exploited in the context of Whitehead’s algorithm there. In this case $\nu_{A}$ already has the ”strict minimality” properties similar to those of strictly minimal elements of $F_{N}$. Again see [24] for details.

(2) Let $\Gamma$ be a simplicial chart on $F_{N}$. Consider a $\Gamma$-based FSMC $\mathcal{X}$ with state set $S=E\Gamma$ and transition probabilities satisfying $p_{\mathcal{X}}(e,e^{\prime})>0$ if and only if $ee^{\prime}$ is a reduced length-2 edge-path in $\Gamma$. Then $\mathcal{X}$ is irreducible, tight. The characteristic current $\nu_{X}$ has full support, and therefore is filling.

(3) Let $\mathcal{X}$ be an $R_{A}$-based FSMC with state set $S=A$ and transition probabilities satisfying $p_{\mathcal{X}}(a_{i},a_{j})>0$ for all $1\leq i,j\leq N$. Then $\mathcal{X}$ is irreducible and tight.The characteristic current $\nu_{\mathcal{X}}$ has the property that for $1\neq v\in F(A)$ we have $\langle v,\nu_{\mathcal{X}}\rangle>0$ if and only if $v$ or $v^{-1}$ is a positive word over $A$. The current $\nu_{\mathcal{X}}$ is filling in $F_{N}$ by Proposition 5.7. We again have $W_{n}=\breve{W}_{n}$ in this case, and moreover, $W_{n}$ is already cyclically reduced because it is a positive word.

(4) Let $\Gamma$ be a ”fan of lollipops”. Namely, $\Gamma$ is a graph with a central vertex $x_{0}$ with $N$ non-closed oriented edges $e_{1},\dots e_{N}$ emanating from $x_{0}$ with $N$ distinct end-vertices $y_{i}=t(e_{i})$, $i=1,\dots,N$. For each of these $e_{i}$ at the vertex $y_{i}$ there is a closed oriented loop edge $f_{i}$ attached, with label $a_{i}\in A$ indicating the marking (so that $f_{i}^{-1}$ is marked $a_{i}^{-1}$). Consider a $\Gamma$-based FSMC $\mathcal{X}$ with state set $S=E\Gamma-\{f_{1}^{-1},\dots,f_{N}^{-1}\}$. The transition probabilities satisfy $p_{\mathcal{X}}(e,e^{\prime})>0$ whenever $e,e^{\prime}\in S$ and $ee^{\prime}$ is a reduced length-2 edge-path in $\Gamma$. Then again $\mathcal{X}$ is irreducible and tight. Moreover, the characteristic current $\nu_{X}\in\mbox{Curr}(F_{N})$ is filling by Proposition 5.7.

In (1), (2), (3) and (4) above, the processes $\widehat{W}_{\mu}$ and $\breve{W}_{\mu}$ (where $\mu$ is any initial distribution on the state set $S$ of $\mathcal{X}$) are adapted to the characteristic current $\nu_{X}$ of the defining tight irreducible FSMC, and $\nu_{X}$ is filling in $F_{N}$. Therefore Theorem 6.11 and Theorem 6.12 apply to $\widehat{W}_{n}$ and $\breve{W}_{n}$ in these cases.