Towards quantitative classification of Cayley automatic groups

TL;DR

This paper introduces a numerical characteristic for Cayley automatic groups, explores its properties, and applies it to various group classes to facilitate their quantitative classification.

Contribution

It formulates and proves properties of a new numerical characteristic for Cayley automatic groups, including its invariance and relationships with other group invariants.

Findings

The numerical characteristic satisfies a fellow traveler property.

It is invariant under finite extensions, direct products, and free products.

The characteristic is studied for nilpotent groups, Heisenberg groups, and groups with exponential growth.

Abstract

In this paper we address the problem of quantitative classification of Cayley automatic groups in terms of a certain numerical characteristic which we earlier introduced for this class of groups. For this numerical characteristic we formulate and prove a fellow traveler property, show its relationship with the Dehn function and prove its invariance with respect to taking finite extension, direct product and free product. We study this characteristic for nilpotent groups with a particular accent on the Heisenberg group, the fundamental groups of torus bundles over the circle and groups of exponential growth.

Equations

$$h(n)=\max \left\{d_A(\pi (w),\psi (w))|w\in L^{⩽n}\right\},$$

$$s(n)=\max \{d_A(\widehat{w_1}(t),\widehat{w_2}(t))|\psi (w_1)g=\psi (w_2),g\in A,t⩽n\}.$$

$$\begin{array}{r}\eta (a,b,c)\equiv \exists r,s_1,s_2,t_1,t_2,t_3\{R_1(r,a)\wedge (R_0(b,s_1)\wedge R_2(s_1,s_2)\wedge R_2(r,s_2))\wedge \\ (R_0(r,t_1)\wedge R_2(t_1,t_2))\wedge (R_2(c,w_0)\wedge R_0(c,t_3)\wedge R_2(t_3,t_2))\}.\end{array}$$

Full text

1 Introduction and Preliminaries

Strings over a finite alphabet appear a natural way to represent elements of a finitely generated group. Following this way Thurston introduced automatic groups which became an important part of geometric group theory [10]. Trying to extend the class of automatic groups, one can either use more powerful computational models (e.g., asynchronous automata, pushdown automata and etc.) or relax the constraint on the correspondence between strings and group elements (for automatic groups this correspondence is given by the canonical map). The latter approach leads to Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov [12]. Utilization of both approaches simultaneously leads further to $\mathcal{C}$–graph automatic groups introduced by Elder and Taback [9]. In this paper we focus only on Cayley automatic groups.

Cayley automatic groups utilize exactly the same computational model as automatic groups, so they preserve some key algorithmic features of automatic groups, but the correspondence between strings and group elements can be arbitrary. Another way to define Cayley automatic groups is to say that they are finitely generated groups for which labeled directed Cayley graphs are automatic (FA–presentable) structures [14, 13, 15]. For a recent survey of the theory of automatic structures we refer the reader to [21]. The class of Cayley automatic groups is essentially wider than the class of automatic groups [12]. Also, Cayley automatic groups include important classes of groups such as nilpotent groups of nilpotency class two, fundamental groups of $3$–manifolds, Baumslag–Solitar groups, restricted wreath products of Cayley automatic groups by the infinite cyclic group, higher rank lamplighter groups [12, 4, 6].

We assume that the reader is familiar with the definitions of finite automata and regular languages (a concise introduction is given in, e.g., [10, Sections 1.1–2]). For a given finite alphabet $\Sigma$ we denote by $\Sigma^{*}$ the set of all finite strings over $\Sigma$ and by $\Sigma_{\diamond}$ the alphabet $\Sigma=\Sigma\cup\{\diamond\}$ (it is assumed that $\diamond\not\in\Sigma$). For any $w\in\Sigma^{*}$, we denote by $|w|$ the length of the string $w$. Let $w_{1},\dots,w_{n}\in\Sigma^{*}$. The convolution $w_{1}\otimes\dots\otimes w_{n}$ is the string of a length $m=\max\{|w_{1}|,\dots,|w_{n}|\}$ over the alphabet ${\Sigma_{\diamond}^{n}}^{\prime}=\Sigma_{\diamond}^{n}\setminus\{(\diamond,\dots,\diamond)\}$ for which the $k$th symbol, $k=1,\dots,m$, is $(\sigma_{1k},\dots,\sigma_{nk})\in{\Sigma_{\diamond}^{n}}^{\prime}$, where $\sigma_{ik}$ is the $k$th symbol of $w_{i}$ if $k\leqslant|w_{i}|$ and $\sigma_{ik}=\diamond$ if $k>|w_{i}|$ for $i=1,\dots,n$. For any relation $R\subseteq\Sigma^{*n}$, we say that $R$ is FA–recognizable (regular) if $\otimes R=\{w_{1}\otimes\dots\otimes w_{n}\,|\,(w_{1},\dots,w_{n})\in R\}$ is a regular language over the alphabet ${\Sigma_{\diamond}^{n}}^{\prime}$. Let $G$ be a finitely generated (f.g.) group and $A\subset G$ be a finite generating set of $G$. Let $A^{-1}$ be the set of the inverses of elements of $A$ and $S=A\cup A^{-1}$. We denote by $\pi:S^{*}\rightarrow G$ the canonical map which maps any given string $w=s_{1}\dots s_{n}\in S^{*}$ to the group element $g=s_{1}\dots s_{n}\in G$.

Definition 1.1**.**

A group $G$ is called Cayley automatic if there exists a bijection $\psi:L\rightarrow G$ between some regular language $L\subseteq\Sigma^{*}$ and the group $G$ for which the binary relation $R_{a}=\left\{\left(\psi^{-1}(g),\psi^{-1}(ga)\right)|g\in G\right\}$ is FA–recognizable for every $a\in A$. Such a bijection $\psi:L\rightarrow G$ is called a Cayley automatic representation of $G$.

In this paper we assume that $\Sigma=S$, unless otherwise stated. This assumption is needed to correctly define the function $h(n)$ in the formula (1.1) below: if $w\in S^{*}$, then $\pi(w)$ is in the group $G$ as well as $\psi(w)$, so one can get the distance $d_{A}(\pi(w),\psi(w))$ between $\pi(w)$ and $\psi(w)$ in the Cayley graph $\Gamma(G,A)$. We recall that for given $g_{1},g_{2}\in G$, the distance $d_{A}(g_{1},g_{2})$ between the elements $g_{1}$ and $g_{2}$ in $G$ with respect to $A$ is the length of a shortest path from $g_{1}$ to $g_{2}$ in the Cayley graph $\Gamma(G,A)$. For a given $g\in G$, we denote by $d_{A}(g)$ the distance $d_{A}(e,g)$, where $e$ is the identity of the group $G$. Since the cardinality of $S$ is at least two, it can be verified that Definition 1.1 (either together with the assumption that $\Sigma=S$ or without it) is equivalent to the original definition of Cayley automatic groups [12, Definition 6.4] (they are also referred as Cayley graph automatic or graph automatic groups in the literature). Furthermore, assuming that $\Sigma=S$ and $\psi=\pi$ in Definition 1.1, one gets the definition of automatic groups; it can be also verified that it is equivalent to the original definition given by Thurston, see [10, Definition 2.3.1]. This observation motivated us to introduce a function (1.1) as a measure of deviation of a given Cayley automatic representation $\psi$ from automatic representations [5]:

[TABLE]

where $L^{\leqslant n}=\{w\in L\,|\,|w|\leqslant n\}$ is the set of strings from $L$ of a length less or equal than $n$. If a group $G$ is Cayley automatic but not automatic, a Cayley automatic representation $\psi$ for which $\psi=\pi$ does not exist. So, in this case, for every Cayley automatic representation $\psi$ of $G$ the function $h(n)$ defined by (1.1) is not identically equal to zero.

We denote by $\mathfrak{F}$ the set of all nondecreasing functions from some interval $[Q,+\infty)\subseteq\mathbb{N}$ to the set of nonnegative real numbers. Clearly, a function $h(n)$ given in (1.1) is in $\mathfrak{F}$. For any given $g,f\in\mathfrak{F}$, we say that $g\preceq f$ ($g$ is coarsely less or equal than $f$) if there exist nonnegative integer $N$ and positive integers $K$ and $M$ for which $g(n)\leqslant Kf(Mn)$ for all $n\geqslant N$. We say that $g\asymp f$ ($g$ is coarsely equal to $f$) if $g\preceq f$ and $f\preceq g$. Similarly, we say that $g\prec f$ ($g$ is coarsely strictly less than $f$) if $g\preceq f$ and $g\not\asymp f$. Clearly, the coarse equality $\asymp$ gives an equivalence relation on $\mathfrak{F}$. In this paper we will be considering functions from $\mathfrak{F}$ up to this equivalence relation.

Any given Cayley automatic group $G$ admits infinitely many Cayley automatic representations $\psi:L\rightarrow G$. So, in general, the problem of finding Cayley automatic representations minimizing coarsely the function (1.1) is nontrivial. In [5, Theorems 11 and 13], we constructed Cayley automatic representations of the Baumslag–Solitar groups $BS(p,q)$, $q>p\geqslant 1$ and the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ which are minimizers of the function (1.1). In both cases the minimum for the function $h(n)$ is the identity function $\mathfrak{i}$: $\mathfrak{i}(n)=n$ for all $n\in\mathbb{N}$. Furthermore, in [5] we introduced classes of Cayley automatic groups $\mathcal{B}_{f}$ as follows. For a given $f\in\mathfrak{F}$, $G\in\mathcal{B}_{f}$ if there exists a Cayley automatic representation $\psi:L\rightarrow G$ for which $h\preceq f$, where $h$ is given by (1.1). In particular, the Baumslag–Solitar groups $BS(p,q)$, $q>p\geqslant 1$ and the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ are in the class $\mathcal{B}_{\mathfrak{i}}$ and they cannot be in any class $\mathcal{B}_{f}$ if $f\prec\mathfrak{i}$.

It is easy to show that the definition of a class $\mathcal{B}_{f}$ does not depend on the choice of generators [5, Proposition 5]. Clearly, $\mathcal{B}_{f}\subseteq\mathcal{B}_{g}$ if $f\preceq g$. Also, for the zero function $\bf{z}$, where ${\bf z}(n)=0$ for all $n\in\mathbb{N}$, the class $\mathcal{B}_{\bf z}$ coincides with the class of automatic groups. In [5, Theorem 8] we proved that there exists no nonautomatic group in any class $\mathcal{B}_{d}$, where $d\in\mathfrak{F}$ is a function bounded from above by some constant; that is, $\mathcal{B}_{d}=\mathcal{B}_{\bf z}$ for any such function $d$. Another group that we considered in [5] was the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$. We showed that $\mathcal{H}_{3}(\mathbb{Z})\in\mathcal{B}_{\mathfrak{e}}$, where $\mathfrak{e}$ is the exponential function: $\mathfrak{e}(n)=\exp(n)$. But a lower bound for $h(n)$ which we could find in the case of $\mathcal{H}_{3}(\mathbb{Z})$ is far from being exponential, it is $\sqrt[3]{n}$ [5, Theorem 15].

For a given $G\in\mathcal{B}_{f}$ we treat $f\in\mathfrak{F}$ as a numerical characteristic of $G$. We especially interested in those $f$ which are sharp lower bounds for (1.1). The fact that the sharp lower bounds can be obtained for some groups sounds promising. Numerical characteristics of groups, e.g. growth functions, Dehn functions, drifts of simple random walks and etc., and relations between them are very important in group theory, see, e.g., [23]. Another motivation to study this numerical characteristic is to address the problem of characterization of Cayley automatic groups; see also [2], where this problem is addressed in terms of numerical characteristics of Turing transducers.

In this paper we continue studying this numerical characteristic of Cayley automatic groups and its relation to other numerical characteristics initiated in [5]. In Section 2 we propose a fellow traveler property for Cayley automatic groups in Theorem 2.1 and show a relation with the Dehn function in Theorem 2.3. The fellow traveler property is well known for automatic groups but its analog for Cayley automatic groups had not been formulated before. In Section 3 we prove invariance of classes $\mathcal{B}_{f}$ under taking finite extension, direct product and free product in Theorems 3.1, 3.2 and 3.3, respectively; in the latter case we require the function $f$ to satisfy a certain inequality.

In Section 4 we show that the semidirect products $\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$, unitriangular matrix groups $UT_{n}(\mathbb{Z})$ and all f.g. nilpotent groups of nilpotency class two are in the class $\mathcal{B}_{\mathfrak{e}}$, see Theorem 4.2. However, this result is obtained from certain Cayley automatic representations of these groups and we do not know whether they are minimizers of the function (1.1) or not. We partly address this issue in Theorem 4.4 by showing that if a virtually nilpotent group $G$ is in a class $\mathcal{B}_{p}$ for some polynomial $p$, then the language $L$ of a Cayley automatic representation $\psi:L\rightarrow G$, for which $h\preceq p$, must be simply starred.

In Section 5 we address the problem of sharp lower bounds of the function (1.1) specifically for the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$. In Theorem 5.1 we show that under a certain condition on a Cayley automatic representation of $\mathcal{H}_{3}(\mathbb{Z})$ the growth of the function (1.1) must be at least exponential. We note that the proof of Theorem 5.1 does not use any knowledge about growth of the Dehn function, which is very often used to show that a given group is not automatic. We believe that Theorem 5.1 can be useful for proposing new approaches to proving nonautomaticity of groups. Section 6 concludes the paper by showing that for any Cayley automatic representation $\psi:L\rightarrow G$ of a group of exponential growth a linear upper bound $d_{A}(\pi(w),\psi(w))\leqslant C|w|$ holds for almost all $w\in L$ in a certain sense, see Theorem 6.1. However, in Remark 6.3 we explain that one should be careful with this simple observation made in Theorem 6.1 by constructing Cayley automatic representations of the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ for which the function (1.1) grows faster than any tower of exponents.

All questions that we posed in [5, §7] remain open. Let us pose an additional question here: is there any Cayley automatic representation of a group of polynomial growth (which is not virtually abelian) or a fundamental group of a $3$–manifold (which is not automatic) for which the function (1.1) is coarsely strictly less than the exponential function $\mathfrak{e}$?

2 **Fellow Traveler Property

and Connection with Dehn Functions**

In this section we formulate a fellow traveler property for Cayley automatic groups and obtain a relation between the Dehn function of a group $G\in\mathcal{B}_{f}$ and a function $f$. For any word $w\in S^{*}$ and nonnegative integer $t$ we put $w(t)$ to be the prefix of $w$ of a length $t$ if $t\leqslant|w|$ and $w$ if $t>|w|$. We denote by $\widehat{w}:[0,\infty)\rightarrow\Gamma(G,A)$ the corresponding path in the Cayley graph $\Gamma(G,A)$: if $t$ is an integer, then $\widehat{w}(t)=\pi(w(t))$ and if $t$ is not an integer, $\widehat{w}(t)$ is obtained by moving along the edge $\left(\widehat{w}(\lfloor t\rfloor),\widehat{w}(\lceil t\rceil)\right)$ with unit speed; we will use only integer values of $t$. Let $\psi:L\rightarrow G$ be any Cayley automatic representation of a group $G$. We denote by $s$ be the following function:

[TABLE]

That is, for every two words $w_{1},w_{2}\in L$ representing neighboring vertices in the Cayley graph $\Gamma(G,A)$ (i.e., for some $g\in A$, $\psi(w_{1})g=\psi(w_{2})$) the distance between $\widehat{w_{1}}(t)$ and $\widehat{w_{2}}(t)$ for all $t\leqslant n$ is bounded from above by $s(n)$. If $G$ is automatic and $\psi$ is an automatic representation of $G$, then $s(n)$ must be a bounded function due to the fellow traveler property for automatic groups [10, Lemma 2.3.2].

Theorem 2.1**.**

Assume that $G\in\mathcal{B}_{f}$ for some nonzero function $f\in\mathfrak{F}$. Then there is a Cayley automatic representation $\psi:L\rightarrow G$ such that for the function $s(n)$ given by (2.1), $s\preceq f$.

**Proof. **Since $G\in\mathcal{B}_{f}$, there exists a Cayley automatic representation $\psi:L\rightarrow G$ such that for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, $h\preceq f$. Let $t,n$ be some nonnegative integers for which $t\leqslant n$ and $w_{1},w_{2}\in L$ be some words representing neighboring vertices in $\Gamma(G,A)$ (i.e., $\psi(w_{1})g=\psi(w_{2})$ for some $g\in A$). The convolution $w_{1}\otimes w_{2}$ is in a regular language $\otimes R_{g}$ accepted by some two–tape synchronous automaton $M_{g}$. Let $T$ be a maximal number of states in the automata $M_{g}$ for all $g\in A$. We assume that $t>T$. If $t\leqslant\max\{|w_{1}|,|w_{2}|\}$, there exist strings $u_{1},v_{1},u_{2},v_{2}$ for which $u_{1}$ and $u_{2}$ are prefixes of $w_{1}(t)$ and $w_{2}(t)$ such that $|u_{1}|,|u_{2}|\geqslant t-T$ and for the strings $w_{1}^{\prime}=u_{1}v_{1}$ and $w_{2}^{\prime}=u_{2}v_{2}$, $|w_{1}^{\prime}|,|w_{2}^{\prime}|\leqslant t$ and the convolution $w_{1}^{\prime}\otimes w_{2}^{\prime}\in\otimes R_{g}$. If $t\geqslant\max\{|w_{1}|,|w_{2}|\}$, then we simply put $u_{1}=w_{1}$, $u_{2}=w_{2}$ and $v_{1}=v_{2}=\epsilon$, where $\epsilon$ is the empty string. We have: $d_{A}(\widehat{w_{1}}(t),\widehat{w_{2}}(t))\leqslant d_{A}(\pi(u_{1}),\pi(u_{2}))+2T\leqslant d_{A}(\pi(w_{1}^{\prime}),\pi(w_{2}^{\prime}))+|v_{1}|+|v_{2}|+2T\leqslant d_{A}(\pi(w_{1}^{\prime}),\pi(w_{2}^{\prime}))+4T$. Moreover, $d_{A}(\pi(w_{1}^{\prime}),\pi(w_{2}^{\prime}))\leqslant d_{A}(\pi(w_{1}^{\prime}),\psi(w_{1}^{\prime}))+d_{A}(\psi(w_{1}^{\prime}),\psi(w_{2}^{\prime}))+d_{A}(\psi(w_{2}^{\prime}),\pi(w_{2}^{\prime}))\leqslant h(|w_{1}^{\prime}|)+1+h(|w_{2}^{\prime}|)\leqslant 2h(t)+1$. Therefore, $d_{A}(\widehat{w_{1}}(t),\widehat{w_{2}}(t))\leqslant 2h(t)+4T+1\leqslant 2h(n)+4T+1$. If $t\leqslant T$, $d_{A}(\widehat{w_{1}}(t),\widehat{w_{2}}(t))$ can be bounded from above by $2T$. Since $h\preceq f$ and $f$ is a nonzero function, then $s\preceq f$. $\Box$

Remark 2.2**.**

Clearly, we have $d_{A}(\widehat{w_{1}}(t),\widehat{w_{2}}(t))\leqslant d_{A}(\widehat{w_{1}}(t))+d_{A}(\widehat{w_{2}}(t))\leqslant 2t$. Therefore, $s\preceq\mathfrak{i}$ for any function $s$ given by (2.1). So, Theorem 2.1 is of interest if $f\prec\mathfrak{i}$. It is not known whether there exists any Cayley automatic group in a class $\mathcal{B}_{f}$, for $f\prec\mathfrak{i}$, which is not automatic. If such groups do not exist, Theorem 2.1 might be a first step to prove it. At least, Theorem 2.1 can serve as an argument to prove that a given group $G\notin\mathcal{B}_{f}$ for some $f\prec\mathfrak{i}$.

Let $G$ be a group $G=\langle A|R\rangle$ defined by a finite set of generators $A$ and a finite set of relators $R$. Let $S=A\cup A^{-1}$. The Dehn function $D(n)$ of $G$ given by $A$ and $R$ is defined as $D(n)=\max\{\mathrm{Area}(w)|w\in S^{\leqslant n}\wedge\pi(w)=e\}$, where $\mathrm{Area}(w)$ is the minimal integer $k$ for which $w=\prod_{i=1}^{k}v_{i}r_{i}^{\pm 1}v_{i}^{-1}$, $r_{i}\in R$, in the free group $F(A)$. Let us assume that $G\in\mathcal{B}_{f}$ for some nonzero function $f\in\mathfrak{F}$. Theorem 2.3 and Corollary 2.4 below extend the results we obtained in [5, Theorems 11 and 15].

Theorem 2.3**.**

Assume that we are given two functions $p,q\in\mathfrak{F}$ for which $p(n)\preceq D(n)\preceq q(n)$. Then $p(n)\leqslant Cn^{2}q(Kf(Mn))$ for all $n\geqslant N$ for some constants $C,K,M$ and $N$. In particular, if $p=q=n^{d}$ for some $d>2$, then $n^{\frac{d-2}{d}}\preceq f$. If $p=q=\mathfrak{e}$, then $\mathfrak{i}\preceq f$.

**Proof. **Let $\psi:L\rightarrow G$ be a Cayley automatic representation of $G$ such that for the function $h(n)=\max\{d_{A}(\psi(w),\pi(w))|w\in L^{\leqslant n}\}$, $h\preceq f$. Let $w=a_{1}\dots a_{n}\in S^{*}$ be a word representing the identity in $G$, where $a_{i}\in S$. For a given $j=1,\dots,n-1$, we put $g_{j}=a_{1}\dots a_{j}$ and $g_{0}=g_{n}=e$. We first divide a loop given by the word $w$ into $n$ subloops as follows. For any $i=0,\dots,n-1$ let $u_{i}\in S^{*}$ be the following concatenation of words: $u_{i}=\eta_{i}\xi_{i}a_{i+1}\xi_{i+1}^{R}\eta_{i+1}^{R}$, where $\eta_{i}=\psi^{-1}(g_{i})$, $\xi_{i}$ is some fixed word traversing a shortest path from $\pi(\eta_{i})$ to $g_{i}$, $\xi_{i+1}^{R}$ and $\eta_{i+1}^{R}$ are the inverses of $\xi_{i+1}$ and $\eta_{i+1}$, respectively; e.g., if $\xi=abbc^{-1}a^{-1}$, then $\xi^{R}=acb^{-1}b^{-1}a^{-1}$. Clearly $\pi(u_{i})=e$, so we obtain a loop.

By the bounded difference lemma (see, e.g., [12, Lemma 14.1]), the length of each string $\eta_{i}$ is bounded by $Cn$ for some constant $C$. Then each of the subloops given by $u_{i}$, $i=0,\dots,n-1$ we divide into at most $Cn$ smaller subloops as follows. For every $1\leqslant j\leqslant\max\{|\eta_{i}|,|\eta_{i+1}|\}$ we construct a loop starting at the point $\widehat{\eta_{i}}(j-1)$ as follows. For $1\leqslant j<\max\{|\eta_{i}|,|\eta_{i+1}|\}$ the loop defined by the word $v_{ij}=p_{ij}\zeta_{ij}p_{(i+1)j}^{R}\zeta_{i(j-1)}^{R}$, where $p_{ij}$ is the string for which $\eta_{i}(j)=\eta_{i}(j-1)p_{ij}$ (so $p_{ij}$ is either a single–letter string or the empty string) and $\zeta_{ij}$ is some word traversing a shortest path from $\widehat{\eta_{i}}(j)$ to $\widehat{\eta_{i+1}}(j)$; clearly, the length of this loop is bounded by $s(j)+s(j-1)+2$, where $s$ is the function given (2.1). For $j=\max\{|u_{i}|,|u_{i+1}|\}$ the loop is defined by the word $v_{ij}=p_{ij}\xi_{i}a_{i+1}\xi_{i+1}^{R}p_{(i+1)j}^{R}\zeta_{i(j-1)}^{R}$; the length of this loop is bounded by $(2h(j)+1)+s(j-1)+2$. Let $\ell^{\prime}(k)=\max\{2s(k)+2,2h(k)+s(k)+3\}$ and $\ell(k)=\ell^{\prime}(Ck)$. So, the length of each of these smaller subloops is bounded by $\ell(n)=\ell^{\prime}(Cn)$. By the inequalities $h\preceq f$ and $s\preceq f$ (see Theorem 2.1), we have $\ell\preceq f$. The total number of these smaller subloops is at most $Cn^{2}$. Thus we obtain the inequality $D(n)\leqslant Cn^{2}D(\ell(n))$. Therefore, $D(n)\preceq n^{2}D(\ell(n))$.

From the inequalities $D(n)\preceq n^{2}D(\ell(n))$, $\ell\preceq f$ and $p(n)\preceq D(n)\preceq q(n)$ we obtain that: $p(n)\leqslant C_{1}D(C_{2}n)\leqslant C_{3}n^{2}D(\ell(C_{4}n))\leqslant Cn^{2}q(C_{5}\ell(C_{4}n)))\leqslant Cn^{2}q(Kf(Mn))$ for all $n\geqslant N$ for some constants $C,K,M,N$ and $C_{i},i=1,\dots,5$. If $p=q=n^{d}$, then $n^{d}\leqslant Cn^{2}(Kf(Mn))^{d}$ for all $n\geqslant N$. Therefore, $n^{\frac{d-2}{d}}\leqslant C^{\frac{1}{d}}Kf(Mn)$ for all $n\geqslant N$, i.e., $n^{\frac{d-2}{d}}\preceq f$. If $p=q=\mathfrak{e}$, then $\exp(n)\leqslant Cn^{2}\exp(Kf(Mn))$ for all $n\geqslant N$. Therefore, $n\leqslant\log C+2\log n+Kf(Mn)$ for all $n\geqslant N$, which implies that $\mathfrak{i}\preceq f$. $\Box$

Corollary 2.4**.**

For a given function $f\in\mathfrak{F}$ we have:

• •

if the Baumslag–Solitar group $BS(p,q)\in\mathcal{B}_{f}$ for some $q>p\geqslant 1$, then $\mathfrak{i}\preceq f$;

• •

if the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})\in\mathcal{B}_{f}$, then $\sqrt[3]{n}\preceq f$;

• •

if the group $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\in\mathcal{B}_{f}$ for a matrix $A\in GL(2,\mathbb{Z})$ with two real eigenvalues not equal to $\pm 1$, then $\mathfrak{i}\preceq f$.

**Proof. **This follows from Theorem 2.3 and the facts that for the groups $BS(p,q)$, $1\leqslant p<q$, $\mathcal{H}_{3}(\mathbb{Z})$ and $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$, for a matrix $A\in GL(2,\mathbb{Z})$ with two real eigenvalues not equal to $\pm 1$, the Dehn functions are exponential, cubic and exponential, respectively (see [8] and, e.g., [10, §7.4–§8.1]). $\Box$

Remark 2.5**.**

We recall that the groups $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ are the fundamental groups of $3$–manifolds which are $2$–dimensional torus bundles over the circle. The Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$ is isomorphic $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ for some unipotent matrix $A$; see also Section 5.

Remark 2.6**.**

The examples of Dehn functions for Cayley automatic groups, which are known to us, are quadratic (e.g, for the higher Heisenberg groups $\mathcal{H}_{2k+1}(\mathbb{Z}),k>1$), cubic (e.g., for the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$), $n^{d}$ for any integer $d>3$ (e.g., for some semidirect products $\mathbb{Z}^{m}\rtimes_{A}\mathbb{Z}$, see [7, 8]), and the exponential function $\mathfrak{e}$ (e.g., for the Baumslag–Solitar groups $BS(p,q)$, $1\leqslant p<q$).

3 **Finite Extensions,

Direct Products,

Free Products**

In this section we show that classes $\mathcal{B}_{f}$ are invariant with respect to taking finite extension, direct product and free product. For the latter case we require that $f$ satisfies the inequality $f(x)+f(y)\leqslant f(x+y)$ for all $x,y\geqslant n_{0}$, where $n_{0}$ is some constant. Let $H$ be a subgroup of finite index in a f.g. group $G$. It is known that if $H$ is automatic, then $G$ is automatic. Moreover, by [12, Theorem 10.1], if $H$ is Cayley automatic, then $G$ is Cayley automatic111A complete analog of [10, Theorem 4.1.4] for automatic groups, claiming that a subgroup $H$ of finite index of a group $G$ is automatic iff $G$ is automatic, is not known for Cayley automatic groups. We remark that in the original [12, Theorem 10.1] the assumption that $H$ is a normal subgroup of $G$ can be omitted; see, e.g., [3, Theorem 2.2.4]..

Theorem 3.1**.**

Let $H$ be a subgroup of finite index of a group $G$. If $H\in\mathcal{B}_{f}$, then $G\in\mathcal{B}_{f}$.

**Proof. **Let us fix a finite set of generators of $H$: $A_{1}=\{h_{1},\dots,h_{n}\}$, and a set of unique representatives of the right cosets $Hg$ of the subgroup $H$ in $G$, where $g\notin H$: $A_{2}=\{k_{1},\dots,k_{m}\}$. We put $S_{1}=A_{1}\cup A_{1}^{-1}$. Since $H\in\mathcal{B}_{f}$, there exist a Cayley automatic representation $\psi_{1}:L_{1}\rightarrow H$, $L_{1}\subseteq S_{1}^{*}$ such that, for the function $h_{1}(n)=\max\{d_{A_{1}}(\pi(u),\psi_{1}(u))\,|\,u\in L_{1}^{\leqslant n}\}$, $h_{1}(n)\preceq f(n)$. Let $L_{2}$ be the finite language consisting of $m$ single–letter strings $k_{1},\dots,k_{m}$ and the empty sting $\epsilon$. We put $\psi_{2}$ to be the natural embedding of these strings into the group $G$: a string $k_{i}$ maps to the group element $k_{i}$ and the empty string $\epsilon$ maps to the identity of the group $G$. We put $L$ to be the concatenation of $L_{1}$ and $L_{2}$. Clearly, $L\subseteq S^{*}$, where $S=A\cup A^{-1}$ and $A=A_{1}\cup A_{2}$. Now, we define the map $\psi:L\rightarrow G$ as follows. Let $w=uv\in L$, where $u\in L_{1}$ and $v\in L_{2}$. We put $\psi(w):=\psi_{1}(u)\psi_{2}(v)$. It is easy to verify that the constructed map $\psi$ is a Cayley automatic representation of the group $G$ (see [12, Theorem 10.1]). Furthermore, $d_{A}(\pi(w),\psi(w))\leqslant d_{A}(\pi(w),\pi(u))+d_{A}(\pi(u),\psi(u))+d_{A}(\psi(u),\psi(w))\leqslant 1+h_{1}(|u|)+1\leqslant h_{1}(|w|)+2$. This immediately implies that for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))\,|\,w\in L^{\leqslant n}\}$, $h\preceq h_{1}$. Therefore, $h\preceq f$. $\Box$

It is known that the direct product of two automatic groups is automatic. The direct product of Cayley automatic groups is also Cayley automatic [12, Corollary 10.4].

Theorem 3.2**.**

If $G_{1},G_{2}\in\mathcal{B}_{f}$, then $G_{1}\times G_{2}\in\mathcal{B}_{f}$.

**Proof. **Let $A_{1}$ and $A_{2}$ be some sets of generators of the groups $G_{1}$ and $G_{2}$ for which $A_{1}\cap A_{2}=\varnothing$; we put $S_{1}=A_{1}\cup A_{1}^{-1}$ and $S_{2}=A_{2}\cup A_{2}^{-1}$. Since $G_{1},G_{2}\in\mathcal{B}_{f}$, there exist Cayley automatic representation $\psi_{1}:L_{1}\rightarrow G_{1}$ and $\psi_{2}:L_{2}\rightarrow G_{2}$ for which the functions $h_{1}(n)=\max\{d_{A_{1}}(\pi(w),\psi_{1}(w))\,|\,w\in L_{1}^{\leqslant n}\}$ and $h_{2}(n)=\max\{d_{A_{2}}(\pi(w),\psi_{2}(w))\,|\,w\in L_{2}^{\leqslant n}\}$ satisfy the inequalities $h_{1}\preceq f$ and $h_{2}\preceq f$, where $L_{1}\subseteq S_{1}^{*}$ and $L_{2}\subseteq S_{2}^{*}$.

Let $L=L_{1}L_{2}$. We construct the map $\psi:L\rightarrow G_{1}\times G_{2}$ as follows. For a given $w=uv$, where $u\in L_{1}$ and $v\in L_{2}$, we put $\psi(w)=(\psi_{1}(u),\psi_{2}(v))\in G_{1}\times G_{2}$. It is easy to verify that the constructed map $\psi$ provides a Cayley automatic representation of $G_{1}\times G_{2}$. The groups $G_{1}$ and $G_{2}$ are naturally embedded in $G_{1}\times G_{2}$, so we have $\pi(w)=\pi(u)\pi(v)=(\pi(u),\pi(v))\in G_{1}\times G_{2}$. Therefore, $d_{A}(\pi(w),\psi(w))\leqslant d_{A}(\pi(u),\psi_{1}(u))+d_{A}(\pi(v),\psi_{2}(v))\leqslant h_{1}(|u|)+h_{2}(|v|)\leqslant h_{1}(|w|)+h_{2}(|w|)=s(|w|)$, where $s(n)=h_{1}(n)+h_{2}(n)$ for all $n\in\mathrm{dom}\,h_{1}\cap\mathrm{dom}\,h_{2}$. Clearly, the inequalities $h_{1}\preceq f$ and $h_{2}\preceq f$ imply that $s\preceq f$. Therefore, for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, we have $h\preceq f$. $\Box$

It is known that the free product of automatic groups is automatic. Therefore, if $G_{1},G_{2}\in\mathcal{B}_{d}$, then $G_{1}\star G_{2}\in\mathcal{B}_{d}$, where $d$ is a bounded function (recall that in this case, by [5, Theorem 8], $\mathcal{B}_{d}$ is the class of automatic groups). Moreover, the free product of Cayley automatic groups is Cayley automatic [12, Theorem 10.8]. In the following theorem we consider the case when $G_{1},G_{2}\in\mathcal{B}_{f}$ for some unbounded function $f\in\mathfrak{F}$.

Theorem 3.3**.**

*Let $f\in\mathfrak{F}$ be a function for which $f(x)+f(y)\leqslant f(x+y)$ for all $x,y\geqslant n_{0}$, where $n_{0}$ is a constant. If $G_{1},G_{2}\in\mathcal{B}_{f}$, then $G_{1}\star G_{2}\in\mathcal{B}_{f}$. *

**Proof. **For initial settings we use the same notation as in the first paragraph of the proof of Theorem 3.2. Without loss of generality we may assume that the empty word $\epsilon\in L_{1},L_{2}$, and $\psi_{1}(\epsilon)$ and $\psi_{2}(\epsilon)$ are the identities in the groups $G_{1}$ and $G_{2}$, respectively. We put $L_{1}^{\prime}=L_{1}\setminus\{\epsilon\}$ and $L_{2}^{\prime}=L_{2}\setminus\{\epsilon\}$. Let $A=A_{1}\cup A_{2}$. Let $L$ be defined by the following regular expression $L=(L_{1}^{\prime}L_{2}^{\prime})^{*}\vee(L_{1}^{\prime}L_{2}^{\prime})^{*}L_{1}^{\prime}\vee(L_{2}^{\prime}L_{1}^{\prime})^{*}\vee(L_{2}^{\prime}L_{1}^{\prime})^{*}L_{2}^{\prime}\vee\epsilon$. That is, $L$ is the regular language consisting of the empty string $\epsilon$ and the strings of the form $u_{1}\dots u_{k}$, where each substring $u_{i}$, $i=1,\dots,k$ either $u_{i}\in L_{1}^{\prime}$ or $u_{i}\in L_{2}^{\prime}$, and no consecutive strings $u_{i},u_{i+1}$ are elements of the same language $L_{1}^{\prime}$ or $L_{2}^{\prime}$. Let us construct the map $\psi:L\rightarrow G_{1}\star G_{2}$ as follows: $\psi(\epsilon)=e$ and $\psi(u_{1}\dots u_{k})=\psi(u_{1})\dots\psi(u_{k})$, where for each $u_{i}$, $i=1,\dots,k$, $\psi(u_{i})=\psi_{1}(u_{i})$ or $\psi(u_{i})=\psi_{2}(u_{i})$ if $u_{i}\in L_{1}^{\prime}$ or $u_{i}\in L_{2}^{\prime}$, respectively. It is easy to verify that the constructed map $\psi$ provides a Cayley automatic representation of $G_{1}\star G_{2}$ (see also [12, Theorem 10.8]).

Now, let $w=u_{1}\dots u_{k}\in L$. Then, $d_{A}(\pi(w),\psi(w))\leqslant d_{A}(\pi(w))+d_{A}(\psi(w))\leqslant|w|+\sum_{i=1}^{k}d_{A}(\psi(u_{i}))$. For each $u_{i}$, $i=1,\dots,k$, we have $d_{A}(\psi(u_{i}))\leqslant d_{A}(\pi(u_{i}))+d_{A}(\pi(u_{i}),\psi(u_{i}))\leqslant|u_{i}|+Kf(M|u_{i}|)$, if $|u_{i}|\geqslant N$ for some positive integer constants $K,M$ and $N$; here we also assume that $MN\geqslant n_{0}$. For all $|u_{i}|<N$ we can bound $d_{A}(\psi(u_{i}))$ from above by some constant $C$ since there exist only finitely many such $u_{i}$; we also assume that $C\geqslant 1$. Therefore, by the assumption that $f(x)+f(y)\leqslant f(x+y)$ for all $x,y\geqslant n_{0}$, we obtain $\sum_{i=1}^{k}d_{A}(\psi(u_{i}))\leqslant C|w|+Kf(M|w|)$. Thus, $d_{A}(\pi(w),\psi(w))\leqslant(C+1)|w|+Kf(M|w|)$ for all $w\in L$. We note that the inequality $f(x)+f(y)\leqslant f(x+y)$ for all $x,y\geqslant n_{0}$ implies that $\mathfrak{i}\preceq f$, unless $f$ is identically equal to zero. So, for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, we have $h\preceq f$. $\Box$

Corollary 3.4**.**

If $G_{1},G_{2}\in\mathcal{B}_{\mathfrak{i}}$ or $G_{1},G_{2}\in\mathcal{B}_{\mathfrak{e}}$, then $G_{1}\star G_{2}$ is also in the class $\mathcal{B}_{\mathfrak{i}}$ or $\mathcal{B}_{\mathfrak{e}}$, respectively.

**Proof. **It is enough to notice that for the functions $f=\mathfrak{i}$ and $f=\mathfrak{e}$, the inequality $f(x)+f(y)\leqslant f(x+y)$ holds for all $x,y\geqslant 1$. $\Box$

4 **Nilpotent Groups and

Fundamental

Groups of $n$–dimensional Torus Bundles over

The Circle**

In this section we show that some classes of nilpotent groups and the fundamental groups of $n$–dimensional torus bundles over the circle are in the class $\mathcal{B}_{\mathfrak{e}}$. In the second half of the section we address the problem of finding sharp lower bounds of the function (1.1) for virtually nilpotent groups. Before we proceed with the main result of the section let us prove the following technical lemma which is needed, in particular, for the proof of Theorem 4.2. Let $\varphi:L_{\Sigma}\rightarrow G$ be a Cayley automatic representation of $G$, where $L_{\Sigma}\subseteq\Sigma^{*}$ now is a regular language over some alphabet $\Sigma^{*}$ (here we do not assume that $\Sigma=S$). We denote by $h_{\varphi}$ the function $h_{\varphi}(n)=\max\{d_{A}(\varphi(w))|w\in L^{\leqslant n}\}$.

Lemma 4.1**.**

Suppose that $h_{\varphi}\preceq f$ for some function $f\in\mathfrak{F}$. Then $G\in\mathcal{B}_{\tilde{f}}$, where $\tilde{f}=f+\mathfrak{i}$.

**Proof. **For every $\sigma\in\Sigma$ let us choose a string $w_{\sigma}\in S^{*}$ such that the lengths $|w_{\sigma}|$ are equal to some constant $\ell$ for all $\sigma\in\Sigma$. Then we define a monoid homomorphism $\xi:\Sigma^{*}\rightarrow S^{*}$ as follows: $\xi(\sigma_{1}\dots\sigma_{k})=w_{\sigma_{1}}\dots w_{\sigma_{k}}$. We define $L=\xi(L_{\Sigma})$ and $\psi=\varphi\circ\xi^{-1}:L\rightarrow G$. Clearly, $\psi:L\rightarrow G$ is a Cayley automatic representation of $G$. Moreover, for any $w\in L$ we have $d_{A}(\pi(w),\psi(w))\leqslant d_{A}(\pi(w))+d_{A}(\psi(w))=|w|+d_{A}(\varphi\circ\xi^{-1}(w))\leqslant|w|+h_{\varphi}(|\xi^{-1}(w)|)=|w|+h_{\varphi}\left(\frac{1}{\ell}|w|\right)\leqslant|w|+h_{\varphi}(|w|)$. Therefore, for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, we clearly have $h\preceq\tilde{f}$. $\Box$

Theorem 4.2**.**

The following groups are all in the class $\mathcal{B}_{\mathfrak{e}}$:

• •

fundamental groups of $n$–dimensional torus bundles over the circle

$\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$,

• •

unitriangular matrices $UT_{n}(\mathbb{Z})$,

• •

f.g. nilpotent groups of nilpotency class $2$.

**Proof. **Let $\beta$ be a representation of $\mathbb{Z}$ for which every $z\in\mathbb{Z}$ is represented as a signed binary number. Let $\gamma$ be a representation of $\mathbb{Z}$ for which every $y\in\mathbb{Z}$ is represented as the concatenation of $|y|$ identical single–letter strings; for positive and negative integers we use different letters. See also the representation of the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$ that we constructed in [5, Section 6]. For any given $\overline{z}=(z_{1},\dots,z_{n})\in\mathbb{Z}^{n}$ we represent it as the convolution $v=w_{1}\otimes\dots\otimes w_{n}$, where $w_{i}=\beta^{-1}(z_{i})$, $i=1,\dots,n$. Then we represent an element $g=(y,\overline{z})\in\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$ as the concatenation $w=uv$, where $u=\gamma^{-1}(y)$. By [12, Theorem 10.3], it provides a Cayley automatic representation $\varphi$ of $\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$. In the group $\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$ the element $g=(y,\overline{z})$ is equal to the product $g=(y,\overline{0})\cdot(0,\overline{z})$, where [math] is the identity of $\mathbb{Z}$ and $\overline{0}$ is the identity of $\mathbb{Z}^{n}$. It is easy to see now that the condition of Lemma 4.1 is satisfied for the representation $\varphi$, the function $f=\mathfrak{e}$ and a natural set of generators $(1,\overline{0})$ and $(0,\overline{e}_{i})$, $i=1,\dots,n$, where $\overline{e}_{i}\in\mathbb{Z}^{n}$ has the $j$th element equal to $\delta_{ij}$, $j=1,\dots,n$. Therefore, $\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}\in\mathcal{B}_{\mathfrak{e}}$.

Any element $g$ of the unitriangular matrix group $UT_{n}(\mathbb{Z})$ is given by a $n\times n$ matrix $M$ with all elements below the main diagonal equal to [math] and all elements of the main diagonal equal to $1$. Let $m_{ij}\in\mathbb{Z}$, $i<j$ be the element of $M$ in row $i$ and column $j$. We denote by $t_{ij}\in UT_{n}(\mathbb{Z})$ the transvection given by a $n\times n$ matrix with all elements on the main diagonal and the element in row $i$ and column $j$ equal to $1$ and all other elements equal to [math]. In the group $UT_{n}(\mathbb{Z})$ the element $g$ is equal to the product of transvections $g=t_{1n}^{m_{1n}}\dots t_{(n-1)n}^{m_{(n-1)n}}\dots t_{13}^{m_{13}}t_{23}^{m_{23}}t_{12}^{m_{12}}$. We represent $g$ as the convolution $s_{12}\otimes\dots\otimes s_{(n-1)n}$, where $s_{ij}=\beta^{-1}(m_{ij})$, $1\leqslant i<j\leqslant n$. Clearly, the condition of Lemma 4.1 is satisfied for this representation, the function $f=\mathfrak{e}$ and the set of generators $\{t_{ij}|1\leqslant i<j\leqslant n\}$. Therefore, $UT_{n}(\mathbb{Z})\in\mathcal{B}_{\mathfrak{e}}$.

It is known that for every f.g. nilpotent group its torsion subgroup is finite. Moreover, every f.g. nilpotent group is residually finite. Therefore, every f.g. nilpotent group has a torsion–free subgroup of finite index. So, by Theorem 3.1, it is enough for us to show that any given torsion–free f.g. nilpotent group $G$ of nilpotency class $2$ is in $\mathcal{B}_{\mathfrak{e}}$. In [12, Theorem 12.4] the authors used Mal’cev coordinates to construct Cayley automatic representation of the group $G$. Below we use their representation to show that $G\in\mathcal{B}_{\mathfrak{e}}$. Let $\overline{a}=(a_{1},\dots,a_{n})\in G^{n}$ be any Mal’cev basis for $G$ associated with the upper central series of $G$. We recall that the factors of the upper central series of a torsion–free nilpotent group are torsion–free. So, for any given $g\in G$, we have a unique presentation of $g$ in $G$ as a product: $g=a_{1}^{k_{1}}\dots a_{n}^{k_{n}}$, where $(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ is a tuple of the Mal’cev coordinates of $g$ with respect to the basis $\overline{a}$. We represent $g$ as the convolution $s_{1}\otimes\dots\otimes s_{n}$, where $s_{i}=\beta^{-1}(k_{i})$, $i=1,\dots,n$. The condition of Lemma 4.1 is satisfied for this representation, the function $f=\mathfrak{e}$ and the set of generators $\{a_{1},\dots,a_{n}\}$. Thus, $G\in\mathcal{B}_{\mathfrak{e}}$. $\Box$

Can any of the groups from Theorem 4.2 be in the class $\mathcal{B}_{f}$ for some $f\prec\mathfrak{e}$? The greatest lower bound for the function $f$ that we can obtain from Theorem 2.3 is $\mathfrak{i}$, see, e.g., Corollary 2.4. However, for some groups, e.g. the higher Heisenberg groups $\mathcal{H}_{2k+1}$, $k>1$, Theorem 2.3 does not give any lower bound (recall that they are nilpotent groups of nilpotency class $2$ and their Dehn functions are quadratic). Thurston proved that automatic nilpotent groups must be virtually abelian (see, e.g., [10, Theorem 8.2.8]). So, by [5, Theorem 8], for any class $\mathcal{B}_{f}$ containing a Cayley automatic nilpotent group (which is not virtually abelian) the function $f$ must be unbounded. Moreover, while for the Baumslag–Solitar groups $BS(p,q)$, $q>p\geqslant 1$ and the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ we obtain the sharp lower bounds [5, Theorem 11 and 13], we do not know whether the lower bounds, which we can obtain from Theorem 2.3 for other groups mentioned in this paper, are sharp. To address this issue we make a simple observation in Theorem 4.4 that might, potentially, be useful in the search for the sharp lower bounds for virtually nilpotent groups. Furthermore, in Theorem 5.1 we show that, for the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$, the exponential function $\mathfrak{e}$ is a lower bound of the function (1.1), if one puts some additional constraints on a Cayley automatic representation $\psi$. We recall that a regular language $L$ is called simply starred if a regular expression for $L$ is of the form: $R_{1}\vee\dots\vee R_{I},$ where $R_{i}=v_{i,0}u_{i,1}^{*}v_{i,1}\dots v_{i,P_{i-1}}u_{i,P_{i}}^{*}v_{i,P_{i}}$ for $i=1,\dots,I$. We have the following proposition.

Proposition 4.3** (polynomial growth condition).**

A regular language $L$ has polynomial growth if it is simply starred and exponential growth otherwise.

**Proof. **For the proof see, e.g., [10, Theorem 8.2.8]. $\Box$

Let $\psi:L\rightarrow G$ be a Cayley automatic representation of a virtually nilpotent group $G$; as usual, $L\subseteq(A\cup A^{-1})^{*}$ for some set of generators $A\subset G$. Let $h$ be the function defined by (1.1) corresponding to the representation $\psi$.

Theorem 4.4**.**

Suppose that $h\preceq p$ for some polynomial $p$. Then the language $L$ is simply starred.

**Proof. **For any given $w\in L^{\leqslant n}$ we have $d_{A}(\psi(w))\leqslant d_{A}(\pi(w))+d_{A}(\pi(w),\psi(w))$ $\leqslant n+h(n)$. Therefore, since $h\preceq p$, there exists a polynomial $q$ for which $\psi(w)$ must be in the ball $B_{q(n)}\subset G$ of radius $q(n)$. Recall that a growth function of any virtually nilpotent group is bounded by a polynomial. Therefore, the cardinality of $B_{q(n)}$ must be bounded by $r(n)$ for some polynomial $r$ so the cardinality of the set $L^{\leqslant n}$. By Proposition 4.3 we obtain the statement of the theorem. $\Box$

5 **In The Search for

Alternative Approaches to Proving Nonautomaticity **

In this section we focus on the problem of finding a sharp lower bound of the function (1.1) for the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$. Another motivation of this section is to propose alternative methods for proving nonautomaticity of groups. Clearly, if a group $G\notin\mathcal{B}_{f}$ for some function $f\in\mathfrak{F}$, then $G$ is not automatic. We already know two ways to show that a group is not in a class $\mathcal{B}_{f}$ if $f\prec f_{0}$ for some nonzero function $f_{0}$ (see Theorem 2.3 and the proof that the lamplighter group is not in the class $\mathcal{B}_{f}$ for any $f\prec\mathfrak{i}$ [5, Theorem 13]). In the first approach we use the Dehn function (when it grows faster than the quadratic function), while in the second approach we implicitly use a fact that the lamplighter group is not finitely presented. However, in both cases one straightforwardly gets nonautomaticity by [10, Theorem 2.3.12]. Is there any alternative method to show that a given group $G$ is not in $\mathcal{B}_{f}$ for some $f\in\mathfrak{F}$? Such a method could potentially provide a new way to prove nonautomaticity. In this part we make a first tiny step in this direction focusing on the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$.

It was first noticed by Sénizergues that the Heisenberg group is not automatic, but its Cayley graph is FA–presentable; also, it was one of the first examples of such groups. Another motivation to focus on $\mathcal{H}_{3}(\mathbb{Z})$ is the ”Heisenberg alternative” – each f.g. group $G$ of polynomial growth is either virtually abelian or $\mathcal{H}_{3}(\mathbb{Z})$ can be embedded into $G$. In [18], Nies and Thomas used this alternative to give a new proof of the theorem that every f.g. FA–presentable group is virtually abelian; this was first proved by Oliver and Thomas in [19]. We recall that $\mathcal{H}_{3}(\mathbb{Z})$ is the group of all matrices of the form: \left(\begin{array}[]{ccc}1&x&z\\ 0&1&y\\ 0&0&1\end{array}\right), where $x,y$ and $z$ are integers; so, every element $g\in\mathcal{H}_{3}(\mathbb{Z})$ corresponds to a triple $(x,y,z)$. We denote by $s,p$ and $q$ the group elements corresponding to the triples $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$, respectively. If $g$ corresponds to a triple $(x,y,z)$, then $gs,gp$ and $gq$ correspond to the triples $(x+1,y,z)$, $(x,y+1,x+z)$ and $(x,y,z+1)$, respectively. We put $A=\{e,s,p,q\}$ and $S=\{e,s,p,q,s^{-1},p^{-1},q^{-1}\}$.

It is straightforward to verify that $\mathcal{H}_{3}(\mathbb{Z})$ is isomorphic to the semidirect product $\mathbb{Z}^{2}\rtimes_{T}\mathbb{Z}$, where T=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right): an isomorphism is given by the following mapping (x,y,z)\mapsto\left(y,\left[\begin{array}[]{c}x\\ z\end{array}\right]\right). We denote by $H$ the normal subgroup of $\mathcal{H}_{3}(\mathbb{Z})$ generated by $s$ and $q$, and by $N$ the cyclic subgroup of $\mathcal{H}_{3}(\mathbb{Z})$ generated by $p$. Clearly, $H\cong\mathbb{Z}^{2}$, $N\cong\mathbb{Z}$ and $\mathcal{H}_{3}(\mathbb{Z})=NH$. We denote by $\varphi$, $p_{1}$ and $p_{2}$ the endomorphisms of the group $H$ given by the matrices T=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right), P_{1}=\left(\begin{array}[]{cc}1&0\\ 0&0\end{array}\right) and P_{2}=\left(\begin{array}[]{cc}0&0\\ 0&1\end{array}\right), respectively. The endomorphisms $p_{1}$ and $p_{2}$ are the projectors of $H$ on the cyclic subgroups generated by $s$ and $q$, respectively. We denote these subgroups by $H_{1}$ and $H_{2}$: $H_{1}=p_{1}(H)=\langle s\rangle$ and $H_{2}=p_{2}(H)=\langle q\rangle$. Let $\psi:L\rightarrow\mathcal{H}_{3}(\mathbb{Z})$ be a Cayley automatic representation of $\mathcal{H}_{3}(\mathbb{Z})$, where $L\subseteq S^{*}$. We denote by $L_{H}$ the language $L_{H}=\psi^{-1}(H)\subset L$ and by $w_{0}$ the string $w_{0}=\psi^{-1}(e)$, where $e$ is the identity of the group $\mathcal{H}_{3}(\mathbb{Z})$. Let $R_{\varphi}=\{\langle w,\psi^{-1}\circ\varphi\circ\psi(w)\rangle\,|\,w\in L_{H}\},R_{p_{1}}=\{\langle w,\psi^{-1}\circ p_{1}\circ\psi(w)\rangle\,|\,w\in L_{H}\},R_{p_{2}}=\{\langle w,\psi^{-1}\circ p_{2}\circ\psi(w)\rangle\,|\,w\in L_{H}\}\subset L_{H}\times L_{H}$ be the binary relations on $L_{H}$ defined by the endomorphisms $\varphi,p_{1},p_{2}$, respectively, and the Cayley automatic representation $\psi$. For a given binary relation $R\subseteq S^{*}\times S^{*}$, we denote by $L_{H}\triangleleft R$ and $R\triangleright L_{H}$ the left– and right–restrictions of $R$ on $L_{H}$: $L_{H}\triangleleft R=\{\langle u,v\rangle\in R\,|\,u\in L_{H}\}$ and $R\triangleright L_{H}=\{\langle u,v\rangle\in R\,|\,v\in L_{H}\}$. We denote by $L_{H_{1}}$ and $L_{H_{2}}$ the languages $L_{H_{1}}=\psi^{-1}(H_{1})\subset L_{H}$ and $L_{H_{2}}=\psi^{-1}(H_{2})\subset L_{H}$.

Theorem 5.1**.**

Assume that there exist some FA–recognizable relations $R_{0}$, $R_{1}$, $R_{2}$ $\subseteq S^{*}\times S^{*}$ for which $L_{H}\triangleleft R_{0}=R_{\varphi}$, $R_{1}\triangleright L_{H_{1}}=R_{p_{1}}$, $L_{H}\triangleleft R_{2}=R_{p_{2}}$ and $R_{2}\triangleright\{w_{0}\}=R_{p_{2}}\triangleright\{w_{0}\}$. Then, for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, $\mathfrak{e}\preceq h$. In particular, $h\npreceq f$ for any $f\prec\mathfrak{e}$.

**Proof. **Let $\eta(a,b,c)$ be the following first–order formula:

[TABLE]

Let us verify that for any $a,b\in L_{H_{1}}$ the formula $\eta(a,b,c)$ is true if and only if $c\in L_{H_{1}}$ and $\psi(a)+\psi(b)=\psi(c)$ in the cyclic group $H_{1}$. Suppose that, for some $a,b\in L_{H_{1}}$, $\eta(a,b,c)$ is true. Let \psi(a)=\left[\begin{array}[]{c}k\\ 0\end{array}\right], \psi(b)=\left[\begin{array}[]{c}\ell\\ 0\end{array}\right] for some $k,\ell\in\mathbb{Z}$. Since $R_{1}(r,a)$ is true and $R_{1}\triangleright L_{H_{1}}=R_{p_{1}}$, then $r\in L_{H}$ and \psi(r)=\left[\begin{array}[]{c}k\\ \star\end{array}\right]. Furthermore, since $R_{0}(b,s_{1})\land R_{2}(s_{1},s_{2})\land R_{2}(r,s_{2})$ is true and $L_{H}\triangleleft R_{0}=R_{\varphi}$, $L_{H}\triangleleft R_{2}=R_{p_{2}}$, then $s_{1}\in L_{H}$, $s_{2}\in L_{H_{2}}$ and \psi(s_{1})=\left[\begin{array}[]{c}\ell\\ \ell\end{array}\right], \psi(s_{2})=\left[\begin{array}[]{c}0\\ \ell\end{array}\right], \psi(r)=\left[\begin{array}[]{c}\star\\ \ell\end{array}\right]. Therefore, \psi(r)=\left[\begin{array}[]{c}k\\ \ell\end{array}\right]. Moreover, since $R_{0}(r,t_{1})\land R_{2}(t_{1},t_{2})$ is true and $L_{H}\triangleleft R_{0}=R_{\varphi}$, $L_{H}\triangleleft R_{2}=R_{p_{2}}$, then \psi(t_{1})=\left[\begin{array}[]{c}k\\ k+\ell\end{array}\right], \psi(t_{2})=\left[\begin{array}[]{c}0\\ k+\ell\end{array}\right]. Finally, since $R_{2}(c,w_{0})\land R_{0}(c,t_{3})\land R_{2}(t_{3},t_{2})$ and $R_{2}\triangleright\{w_{0}\}=R_{p_{2}}\triangleright\{w_{0}\}$, $L_{H}\triangleleft R_{0}=R_{\varphi}$, $L_{H}\triangleleft R_{2}=R_{p_{2}}$, then c=\left[\begin{array}[]{c}m\\ 0\end{array}\right], \psi(t_{3})=\left[\begin{array}[]{c}m\\ m\end{array}\right], \psi(t_{2})=\left[\begin{array}[]{c}0\\ m\end{array}\right]. Thus, $c\in L_{H_{1}}$ and $m=k+\ell$ which implies that $\psi(a)+\psi(b)=\psi(c)$. The reverse is straightforward.

Let $R\subseteq S^{*}\times S^{*}\times S^{*}$ be the relation defined by $\eta$, that is, $R(a,b,c)$ is true iff $\eta(a,b,c)$ is true. Since $R_{0},R_{1},R_{2}$ are FA–recognizable, $R$ is FA–recognizable. Let $(M,\times)$ be a monoid generated by $s$, where $M=\{s^{n}\,|\,n\geqslant 0\}\subset H_{1}$ and $\times$ is the group multiplication in $H_{1}$. Clearly, $(M,\times)\cong(\mathbb{N},+)$. Let $u_{n}=\psi^{-1}(s^{n})\in L$ and $L_{M}=\{u_{n}|n\geqslant 0\}\subset L$. It follows directly from [18, Lemma 6] (this lemma was originally proved in [16] for automatic monoids) that there exist constants $C,N_{0}$ for which $|u_{n}|\leqslant C\log n$ for all $n\geqslant N_{0}$, where $|u_{n}|$ is the length of the string $u_{n}$. It follows from the metric inequalities for the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$, see, e.g., [20, Proposition 1.38], that there exist a constant $C_{1}>0$ for which $d_{A}(s^{n})\geqslant C_{1}n$ for all $n\geqslant 0$. We have: $d_{A}(\pi(u_{n}),\psi(u_{n}))\geqslant d_{A}(\psi(u_{n}))-d_{A}(\pi(u_{n}))\geqslant C_{1}n-|u_{n}|\geqslant C_{1}n-C\log n$ for all $n\geqslant N_{0}$. Therefore, there exist some constants $C_{2}>0$ and $N_{1}\geqslant N_{0}$ for which $d_{A}(\pi(u_{n}),\psi(u_{n}))\geqslant C_{2}n\geqslant C_{2}\exp\left(\frac{1}{C}|u_{n}|\right)$ for all $n\geqslant N_{1}$. Clearly, the set $\{|u_{n}|\,|\,u_{n}\in L_{M}\}\subseteq\mathbb{N}$ is infinite. Moreover, by the finite difference lemma (see, e.g., [12, Lemma 14.1]) , $||u_{n+1}|-|u_{n}||\leqslant D$ for every $n\geqslant 0$ and some constant $D$. Therefore, there exists a constant $D_{0}$ such that for every $j\geqslant D_{0}$ there is $u_{n}\in L_{M}$ for which $j\geqslant|u_{n}|>j-D$. Thus, for the function $h(n)=\max\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$, $\mathfrak{e}\preceq h$. The last statement of the theorem is straightforward. $\Box$

Remark 5.2**.**

We note that the conditions of Theorem 5.1 are clearly satisfied for the Cayley automatic representation of the Heisenberg group $\mathcal{H}_{3}(\mathbb{Z})$ constructed in [5, Section 6]. As for FA–recognizable relation $R_{0}\subset S^{*}\times S^{*}$ for which $L_{H}\triangleleft R_{0}=R_{\varphi}$, it exists if, for example, one additionally requires that the left multiplication by $p^{-1}$ in the group $\mathcal{H}_{3}(\mathbb{Z})$ is FA–recognizable; it follows from the fact that for any $h\in H$: $p^{-1}hp=\varphi(h)$.

6 **Linear Upper Bounds for Almost All

Elements in Groups of Exponential Growth**

In this section we show that for an arbitrary bijection $\psi:L\rightarrow G$ between a language $L\subseteq(A\cup A^{-1})^{*}$ and a group $G$ of exponential growth a linear upper bound $d_{A}\left(\pi\left(\psi^{-1}(g)\right),g\right)\leqslant C|\psi^{-1}(g)|$ holds for almost all $g\in G$ in a certain sense, see Theorem 6.1 and Remark 6.2. However, in the following Remark 6.3 we show how to construct Cayley automatic representations of the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ for which the function (1.1) grows faster than any tower of exponents.

Theorem 6.1**.**

Let us assume that $\psi:L\rightarrow G$ is a Cayley automatic representation of a group $G$ which has exponential growth. Then there exist constants $\lambda_{1},\lambda_{2}>0$ such that for almost all $g\in G$: $\lambda_{1}d_{A}(g)\leqslant|w|\leqslant\lambda_{2}d_{A}(g)$, where $\psi(w)=g$. The term almost all here means that $\lim\limits_{n\rightarrow\infty}\frac{\#Q_{n}}{\#B_{n}}=1$, where $B_{n}=\{g\in G\,|\,d_{A}(g)\leqslant n\}$ is the ball of radius $n$ in $G$ and $Q_{n}\subseteq B_{n}$ is defined as $Q_{n}=\{g\in B_{n}\,|\,\lambda_{1}d_{A}(g)\leqslant|w|\leqslant\lambda_{2}d_{A}(g)\}$. In particular, for every $g\in Q_{n}$, $d_{A}(\pi(w),\psi(w))\leqslant\left(1+\frac{1}{\lambda_{1}}\right)|w|$.

**Proof. **The inequality $|w|\leqslant\lambda_{2}d_{A}(g)$ always holds for some $\lambda_{2}>0$ due to the bounded difference lemma. Since $G$ has exponential growth, there exists $\lambda>1$ for which $\#B_{n}\geqslant\lambda^{n}$ for all $n\geqslant n_{0}$. For a given integer $k>0$ we denote by $R_{k}$ the following finite subset of $G$: $R_{k}=\{g\in G\,|\,g=\psi(w),w\in L^{<k}\}$; where $L^{<k}=\{w\in L\,|\,|w|<k\}$. Since $L\subseteq S^{*}$, $\#R_{k}\leqslant|S|^{k-1}$. We denote by $T_{n,k}$ the set $T_{n,k}=B_{n}\setminus R_{k}$. For every $g\in T_{n,k}$, $|w|\geqslant k$. Therefore, if $\lambda_{1}\leqslant\frac{k}{n}$, for every $g\in T_{n,k}$ we have that $\lambda_{1}d_{A}(g)\leqslant|w|$; so $T_{n,k}\subseteq Q_{n}$. We notice that $\frac{\#T_{n,k}}{\#B_{n}}\geqslant 1-\frac{\#R_{k}}{\#B_{n}}\geqslant 1-\frac{|S|^{k-1}}{\lambda^{n}}$ for all $n\geqslant n_{0}$. So, it is enough to provide $\lambda_{1}$ and a sequence $k_{n},n\geqslant n_{0}$ for which $\lambda_{1}\leqslant\frac{k_{n}}{n}$ for all $n\geqslant n_{0}$ and $\lim\limits_{n\rightarrow\infty}\frac{|S|^{k_{n}-1}}{\lambda^{n}}=0$. We note that $\frac{|S|^{k_{n}-1}}{\lambda^{n}}=\frac{|S|^{k_{n}-1}}{|S|^{(\log_{|S|}\lambda)n}}=\frac{1}{|S|^{(\log_{|S|}\lambda)n-k_{n}+1}}$. Let us put $k_{n}=\lceil\frac{1}{2}(\log_{|S|}\lambda)n\rceil$ for all $n\geqslant n_{0}$ and $\lambda_{1}=\frac{1}{2}(\log_{|S|}\lambda)$. Therefore, $(\log_{|S|}\lambda)n-k_{n}+1\geqslant\frac{1}{2}(\log_{|S|}\lambda)n$, so $\lim\limits_{n\rightarrow\infty}\frac{|S|^{k_{n}-1}}{\lambda^{n}}=0$. Moreover, $\frac{k_{n}}{n}\geqslant\lambda_{1}$ for all $n\geqslant n_{0}$. In order to prove the last inequality, we observe that $d_{A}(\pi(w),\psi(w))\leqslant|w|+d_{A}(g)\leqslant|w|+\frac{1}{\lambda_{1}}|w|$. $\Box$

Remark 6.2**.**

It is easy to see that Theorem 6.1 holds for any bijection $\psi:L\rightarrow G$ such that $||\psi^{-1}(ga)|-|\psi^{-1}(g)||\leqslant C$ for all $g\in G$ and every generator $a\in A$, where $C$ is a constant. Moreover, for the inequality $\lambda_{1}d_{A}(g)\leqslant|w|$ and, accordingly, the inequality $d_{A}(\pi(w),\psi(w))\leqslant\left(1+\frac{1}{\lambda_{1}}\right)|w|$, no assumption is needed – it holds for almost all $g\in G$ for any bijection $\psi$ between a language $L$ and a group $G$ of exponential growth. Since in this paper we focus mainly on Cayley automatic representations of groups, in Theorem 6.1 we assume that $\psi:L\rightarrow G$ is a Cayley automatic representation of $G$.

Remark 6.3**.**

We note that although for any Cayley automatic representation $\psi:L\rightarrow G$ of a group of exponential growth $G$ the inequality $d_{A}(\pi(w),\psi(w))\leqslant C|w|$ holds for some constant $C$ for almost all $\psi(w)=g\in G$ in the sense of Theorem 6.1, it does not hold for all $g\in G$. For example, let us consider the following Cayley automatic representation $\varphi:L_{\Sigma}\rightarrow\mathbb{Z}_{2}\wr\mathbb{Z}$ over the alphabet $\Sigma=\{+,-,0,1,C_{0},C_{1},\#\}$. For any given pair $(f,z)\in\mathbb{Z}_{2}\wr\mathbb{Z}$, we represent it as the string: $u\#f(s)\dots C_{f(z)}\dots f(t)$, where $s$ and $t$ are the minimum and the maximum integers of the set $\{i|f(i)=1\}\cup\{z\}$, $C_{f(z)}$ is $C_{0}$ or $C_{1}$ if $f(z)=0$ or $f(z)=1$, respectively, and the string $u$ is a binary representation of the integer $s$. For example, let us consider a pair $(f,3)$, where $f(1)=1$,$f(2)=1$ and $f(i)=0$ if $i\neq 1,2$, it is represented as the string: $+1\#11C_{0}$. Let us consider a pair $(f,-3)$, where $f(-4)=1$,$f(-3)=1$, $f(-2)=1$, $f(i)=0$ if $i\neq-4,-3,-2$, it is represented as the string: $-100\#1C_{1}1$. We also refer the reader to [3, Example 4.2.1].

One can then convert this representation $\varphi$, in a same way as in Lemma 4.1, into some representation $\psi:L\rightarrow\mathbb{Z}_{2}\wr\mathbb{Z}$ over the alphabet $S=A\cup A^{-1}$, where $A=\{a,t\}$ is the standard set of generators of $\mathbb{Z}_{2}\wr\mathbb{Z}$: $a$ is the nontrivial element of $\mathbb{Z}_{2}$ and $t$ is a generator of $\mathbb{Z}$ (here we treat $\mathbb{Z}_{2}$ and $\mathbb{Z}$ as the subgroups of $\mathbb{Z}_{2}\wr\mathbb{Z}$). Although $\mathbb{Z}_{2}\wr\mathbb{Z}\in\mathcal{B}_{\mathfrak{i}}$ [5, Theorem 13], for the representation $\psi$ the inequality $d_{A}(\pi(w),\psi(w))\leqslant C|w|$ does not hold for all $w\in L$ and any constant $C$. In order to see that, let us consider the representatives $w_{i}=\psi^{-1}(g_{i})$ of the elements $g_{i}=(f_{0},i)\in\mathbb{Z}_{2}\wr\mathbb{Z}$, $i>0$ with respect to $\psi$, where $f_{0}(j)=0$ for all $j\in\mathbb{Z}$. Apparently, $d_{A}(g_{i})=i$ but the function $\ell(i)=|w_{i}|$ grows, coarsely, as $\log i$. So, the function $h(n)=\mathrm{max}\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$ grows at least as fast as the exponential function.

Moreover one can construct a Cayley automatic representation $\psi:L\rightarrow\mathbb{Z}_{2}\wr\mathbb{Z}$ for which the function $h(n)=\mathrm{max}\{d_{A}(\pi(w),\psi(w))|w\in L^{\leqslant n}\}$ grows faster than any tower of exponents $e^{e^{\dots{}^{e}}}$. This follows from the result shown by Frank Stephan:

Theorem 6.4** **(Frank Stephan

[22]).

There exists an automatic representation $\tau:L_{\tau}\rightarrow\mathbb{N}$ of the structure $(\mathbb{N},S)$, where $S$ is the successor function, for which the function $r(n)=\max\{\tau(w)|w\in L_{\tau}^{\leqslant n}\}$ grows faster than any tower of exponents $e^{e^{\dots{}^{e}}}$.

Clearly, one cannot directly generalize Theorem 6.1 for Cayley automatic groups of subexponential growth. Moreover, it simply does not hold for many Cayley automatic groups of subexponential growth – consider, for example, a binary representation of the infinite cyclic group $\mathbb{Z}$. A f.g. group of subexponential growth has either intermediate growth or polynomial growth. Miasnikov and Savchuk constructed a FA–presentable graph of intermediate growth [17]. However, it is still unknown whether there exists any Cayley automatic group of intermediate growth. As for f.g. groups of polynomial growth, due to celebrated Gromov’s theorem [11], any such group is virtually nilpotent.

Acknowledgment

The authors thank Murray Elder, Bakhadyr Khoussainov and Frank Stephan for useful comments.