Word automatic groups of nilpotency class 2

TL;DR

This paper investigates the word automaticity of certain nilpotent groups of class 2 with prime exponent, demonstrating that some are automatic while others are not, and introduces a method for showing automaticity of central extensions.

Contribution

It establishes which nilpotent class 2 groups with prime exponent are word automatic and introduces a new method for proving automaticity of central extensions.

Findings

The infinitely generated free group of this variety is not word automatic.

The infinite extra-special p-group E_p is word automatic.

A new method using co-cycles is introduced for automaticity of central extensions.

Abstract

We consider word automaticity for groups that are nilpotent of class $2$ and have exponent a prime $p$. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special $p$-group $E_p$ is word automatic, as well as an intermediate group $H_p$ which has an infinite centre. In the last section we introduce a method to show automaticity of central extensions of abelian groups via co-cycles.

Equations

$$\begin{array}[]{|c|c|c|c|c|}\hline\cr 1&0&1&\Diamond&\Diamond\\ 1&1&0&0&1\\ 0&0&0&1&1\\ \hline\cr\end{array}.$$

$$\big{[}\underline{\alpha}c,\underline{\beta}d\big{]}=\prod_{r<s\leq n}[x_{r},x_{s}]^{\alpha_{r}\beta_{s}-\alpha_{s}\beta_{r}}$$

$$\underline{\alpha }\cdot \prod _{r<s<n}[x_r,x_s{]}^{v_{r,s}},$$

$$\text{for each }v,w\in V_n\text{ and each }i<p,\text{ one has }w\circ u_n^i\circ [v,u_n]{\le }_{L}z.$$

$$E$$

$$H$$

$$\underline{\alpha }\cdot \underline{\beta }=z^{-{\sum }_{k=1}^{n-1}{\alpha }_k({\sum }_{i=0}^{k-1}{\beta }_i)}\cdot \underline{\alpha +\beta }.$$

$$\underline{\alpha }\cdot \underline{\beta }=\prod _{k=1^{n-1}}{z}_k^{-{\alpha }_k({\sum }_{i<k}{\beta }_i)}\cdot \underline{\alpha +\beta }.$$

$$f(u,v)+f(u+v,w)=f(v,w)+f(u,v+w).$$

$$(u,a)\cdot (v,b):=(u+v,a+b+f(u,v))$$

Full text

1. Introduction

A structure in a finite signature is called word automatic (or FA-presentable) if the domain is a set of strings that can be recognized by a finite automata (FA) over an alphabet $\Sigma$. The atomic relations can be recognized by FA as well, as follows: To check whether an atomic relation holds for elements $a_{1},\ldots,a_{n}$ of the domain, the strings representing these elements are extended to strings $\sigma_{1},\ldots,\sigma_{n}$ of the same length by using a filler symbol $\Diamond$ that is not in $\Sigma$; one requires that a finite automaton recognizes the language over $(\Sigma\cup\{\Diamond\})^{n}$ that consists of the strings obtained by stacking such strings $\sigma_{1},\ldots,\sigma_{n}$ on top of each other.

Word automatic structures were first considered by Hodgson [6, 7] who used them to give a new proof that the first-order theory of $({\mathbb{N}},+)$ is decidable. They were studied in depth by Khoussainov and Nerode [9], the founding paper of the area.

Since finite automata are devices of a very limited computational power, it can be challenging to find nontrivial examples of word automatic structures in a particular class; in some cases, such as for the class of Boolean algebras, it can be shown that only the obvious structures are word automatic [10].

The class of word automatic groups (not to be confused with automatic groups in the sense of Thurston) has good closure properties; for instance, it is closed under finite direct products, and quotients by regular normal subgroups. In the abelian setting is also closed under a certain type of FA-recognizable amalgamation [14]. We note that for any finite group $S$, the direct power $S^{(\omega)}$ is word automatic.

The abelian case. Nies and Semukhin [14] constructed word automatic torsion-free indecomposable abelian groups of rank $2$ and larger. This sets them apart from examples such as ${\mathbb{Z}}\times{\mathbb{Z}}$, which can be considered word automatic in a trivial way. Their examples involved divisibility by more than one prime. It is still open whether a group of this kind can be obtained as a subgroup of $(\mathbb{Z}[1/p])^{n}$ for any $n\geq 2$. Braun and Strüngmann [1], building on methods of Tsankov [16], provided strong restrictions on torsion free abelian groups. In particular, such groups have finite rank.

New examples of word automatic groups. In this paper we study indecomposable word automatic groups in varieties just beyond the abelian. Recall that a group $G$ is nilpotent of class 2 if the law $[[x,y],z]$ holds in $G$. In other words, $G$ is a central extension of an abelian group by another. Fix an odd prime $p$, and let $\mathcal{N}_{2,p}$ denote the variety of groups that are nilpotent of class $2$ and have exponent $p$ (that is, $x^{p}=1$ for each $x$). Note that for each $G\in\mathcal{N}_{2,p}$, the centre $Z(G)$ is an elementary abelian $p$-group, and can thus be seen as a vector space over the field $R$. The same holds for the central quotient $G/Z(G)$.

Let $R=GF(p)$ be the field of $p$ elements. The new examples of word automatic groups are infinitely generated variants of the unitriangular group

$\operatorname{\text{\it UT}}_{3}(R)=\left\{\left(\begin{matrix}1&a&c\\ 0&1&b\\ 0&0&1\end{matrix}\right)\colon\,a,b,c\in R\right\}$.

This is the free group of rank 2 in $\mathcal{N}_{2,p}$. It is generated by $x_{0}=\left(\begin{matrix}1&1&0\\ 0&1&0\\ 0&0&1\end{matrix}\right)$ and $x_{1}=\left(\begin{matrix}1&0&0\\ 0&1&1\\ 0&0&1\end{matrix}\right)$. We have $[x_{0},x_{1}]=z:=\left(\begin{matrix}1&0&1\\ 0&1&0\\ 0&0&1\end{matrix}\right)$. The centre of $\operatorname{\text{\it UT}}_{3}(R)$ is the cyclic group generated by $z$.

We introduce groups $G\in\mathcal{N}_{2,p}$ by varying the definition of $\operatorname{\text{\it UT}}_{3}(R)$. We posit that $G$ have an infinite sequence of distinct generators ${\left\langle{x_{n}}\right\rangle}_{n\in{\mathbb{N}}}$. If the commutators $[x_{i},x_{k}]$, $i<k$ are linearly independent, then $G$ is the free group of infinite rank in $\mathcal{N}_{2,p}$, which we will denote by $F_{\infty}$. We will show that this group is not word automatic. In a nutshell, for any FA-presentation of $F_{\infty}$, the linear independence of the commutators of generators would require $p^{\binom{n}{2}}$ strings of length $O(n)$ to represent the linear combinations of the $[x_{i},x_{k}]$ for $i<k<n$, which is contradictory for large enough $n$.

In contrast, if the commutators $[x_{i},x_{k}]\in Z(G)$ are dependent in a certain strong way, then $G$ is word automatic. The simplest example is a group that we will denote by $E_{p}$, or just $E$ if $p$ is understood: one requires that there is an element $z\neq e$ such that $[x_{i},x_{k}]=z$ for each $i<k$. Thus, the centre is cyclic as in the case of $\operatorname{\text{\it UT}}_{3}(R)$. We note that $E_{p}$ is an extra-special $p$-group in the sense of Higman and Hall: the centre is cyclic of order $p$, equals the derived subgroup, and quotient by the centre is an elementary abelian $p$-group (i.e., a vector space over $GF(p)$). A slightly more complex example is the group $H_{p}$: we require that $[x_{i},x_{k}]=z_{k}$ for $i<k$, where the $z_{k}$ form a basis of the centre.

Abelian subgroups of finite index. Nies and Thomas [15] proved that each finitely generated subgroup of a word automatic group has an abelian subgroup of finite index. This indicates that word automatic groups are close to abelian. It remains open whether each torsion-free word automatic group has an abelian subgroup of finite index [13, Question 4.5]. The groups $E_{p}$ and $H_{p}$ are word automatic torsion groups without abelian subgroups of finite index. The first example of a group of this kind was [15, Example 12], which we will revisit as Example 6.2 below.

The isomorphism problem. The computational complexity of the isomorphism problem is a a good indicator of the complexity of a class of word automatic structures. This problem asks whether two presentations given by finite automata describe isomorphic structures. It was a central topic in Khoussainov, Nies, Rubin and Stephan [10]. They showed that the isomorphism problem is $\Sigma^{1}_{1}$-complete for word automatic graphs, but decidable for word automatic Boolean algebras. Kuske, Liu and Lohrey [11] proved that the isomorphism problem for word automatic equivalence relations is $\Pi^{0}_{1}$-complete. For abelian groups, as well as groups at large, the complexity is unknown ([2, Question 5.1]). To show that the isomorphism problem for a class of word automatic structures is undecidable, one attempts to find constructions for sufficiently complicated structures in the class in order to encode some undecidable problem. In contrast, towards showing its decidability, one attempts to provide restrictions on structures in the class. It is unknown at present whether the isomorphism problem for the class of word automatic groups in $\mathcal{N}_{2,p}$ decidable. The present paper follows both approaches to this, by providing both examples and restrictions for this class.

Some model theory of extra-special $p$-groups. We note that the extra-special $p$-groups $E_{p}$ have appeared in numerous places in the literature. For instance, let us briefly review some model theoretic properties of these groups for $p\neq 2$, due to Felgner [4]. (He denotes these groups by $G(p,\leq)$ where $\leq$ is the usual ordering of $\omega$.) On page 423 he provides a recursive axiom system for the theory of $E_{p}$. It expresses that the group has exponent $p$, and that the centre is cyclic of order $p$ and contains the derived subgroup, which is non-trivial. Furthermore, it expresses that the quotient by the centre is infinite, using an infinite list of axioms. This implies that $E_{p}$ is $\omega$-categorical, since up to isomorphism there is only one countably infinite extra-special group of exponent $p$ as shown in Newman [12].

A group is called pseudofinite if every first-order sentence that holds in it also holds in a finite group (for background on this notion see [8]). Note that for each odd $k$ there is an extra-special group of exponent $p$ and order $p^{k}$. So, any finite set of the axioms can be satisfied in a finite model. Hence $E_{p}$ is pseudo-finite.

2. Preliminaries on word automaticity and on groups in $\mathcal{N}_{2,p}$

Presentations via finite automata: facts and examples

Definition 2.1**.**

One says that a structure $A$ in a finite signature is word automatic (or FA presentable) if the elements of the domain can be represented, possibly ambiguously, by the strings in a regular language $D_{A}$ over an alphabet $\Sigma$ such that the following holds. For each atomic relation of the type $Ra_{1},\ldots,a_{n}$, or $f(a_{1},\ldots,a_{n})=b$, or $a=b$, where $R,f$ are $n$-ary relation, respectively function symbols and $a_{1},\ldots,a_{n},b\in A$, there is a finite automaton that recognizes it in the sense discussed at the beginning of the paper. An FA presentation is a collection of FA as above.

Example 2.2**.**

The structure $({\mathbb{N}},+)$ is word automatic via the usual binary expansion of a natural number. The alphabet is $\Sigma=\{0,1\}$, and the domain consists of the strings ending in $1$, and the empty string. One represents a number $n$ by a binary string $\alpha$, with the least significant digit first. Thus $n=n_{\alpha}=\sum_{i<|\alpha|}\alpha(i)2^{i}$. The empty string denotes [math]. A finite automaton over the alphabet $(\Sigma\cup\{\Diamond\})^{3}$ checks the correctness of the sum $n_{\alpha}+n_{\beta}=n_{\gamma}$ via the carry bit procedure, where the carry bit moves to the right. For instance, if $\alpha=101,\beta=11001$ and $\gamma=00011$, the automaton checks that the sum is correct by accepting the string

[TABLE]

In the definition of FA presentations above, note that for terms $s,t$ the equation $s=t$ is an atomic relation. So, this definition allows for equality to be a nontrivial equivalence relation $E$ on $D_{A}$, which on occasion is useful in defining FA presentations. However, since the length-lexicographical ordering $\leq_{L}$ on $D_{A}$ is FA-recognizable, one can replace $D_{A}$ by the regular set $\{x\in D_{A}\colon\forall y\,[yEx\,\to\,x\leq_{L}y]\}$, and thereby uniquely represent elements by strings.

Given an FA presentation of a structure $\mathcal{A}$ and a formula $\phi$ (possibly with parameters), one can effectively determine a finite automaton recognizing the relation on $\mathcal{A}$ defined by $\phi$. The proof is by induction on $|\phi|$. To deal with existential quantifiers, one uses that for each non-deterministic finite automaton, there is a deterministic one that recognizes the same language. Hence, to show that a group $(G,\circ,{}^{-1})$ is word automatic, it suffices to provide an FA recognizing the binary group operation $\circ$; the unary group operation is definable from it.

Automatic groups in the sense of Epstein and Thurston (see  [3]) are finitely generated by definition. In contrast, the appropriate setting for obtaining interesting word automatic groups is outside the finitely generated, for a finitely generated group is word automatic if and only if it is has an abelian subgroup of finite index. Note that word automatic groups are called finite automata presentable in [13] in order to avoid confounding the two notions.

Preliminaries on nilpotent-2 groups. If a sequence ${\left\langle{x_{i}}\right\rangle}_{i\in{\mathbb{N}}}$ of generators of a group $G$ under discussion is fixed, for a tuple of integers  $\alpha$ we let

$\underline{\alpha}:=\prod_{i<|\alpha|}x_{i}^{\alpha_{i}}$.

For group elements $x,y$, we define the commutator by $[x,y]=x^{-1}y^{-1}xy$. If $G$ is nilpotent of class 2, the commutator induces an alternating bilinear form

$\theta_{G}\colon(G/Z)^{2}\to Z$

where $Z=Z(G)$ and $\theta_{G}(aZ,bZ)=[a,b]$ which is well-defined. Hence, for $|\alpha|=|\beta|=n$ and central elements $c,d$,

[TABLE]

This identity will be used below without mention.

3. The group $F_{\infty}$ is not word automatic

Recall that by $F_{\infty}$ we denote the free group of infinite rank in the variety $\mathcal{N}_{2,p}$ of groups of exponent $p$ and nilpotency class 2. Let ${\left\langle{x_{i}}\right\rangle}_{i\in{\mathbb{N}}}$ be a sequence of free generators, and recall that we write ${\underline{\alpha}}:=\prod_{i<|\alpha|}x_{i}^{\alpha_{i}}$. It is well known ([17]that each element of $F_{\infty}$ has a normal form

[TABLE]

where $|\alpha|=n$ and $0\leq v_{r,s}<p$. (To make this normal form unique, one can require that $\alpha_{n-1}\neq 0\vee\exists r[v_{r,n-1}\neq 0]$; i.e., $n$ is chosen minimal. The central elements are the ones where $\alpha$ is a string of [math]s.)

Theorem 3.1**.**

$F_{\infty}$* is not word automatic.*

Proof.

Assume for a contradiction that $F_{\infty}$ has an FA presentation. It has as a domain a regular set $D\subseteq\Sigma^{*}$ for some alphabet $\Sigma$, and has an FA-recognizable operation $\circ\colon D\times D\to D$ for the binary group operation. As discussed in Section 2, we may assume that the equivalence relation denoting equality in the represented group is equality on $D$. Hence we can identify $F_{\infty}$ with the structure on the domain $D$. As before, by $\leq_{L}$ we denote the length-lexicographical ordering on $D$, which is FA-recognizable as well.

The idea of the proof is to define sequences ${\left\langle{u_{n}}\right\rangle}_{n\in{\mathbb{N}}}$ and ${\left\langle{z_{n}}\right\rangle}_{n\in{\mathbb{N}}}$ of elements of $D$ such that the central elements $[u_{i},u_{k}]$, $i<k<n$, are linearly independent over the field $GF(p)$, and for each $n$, the subgroup generated by $u_{0},\ldots,u_{n}$ is contained in the set of strings $\{x\in D\colon x\leq_{L}z_{n+1}\}$. Furthermore, $z_{n+1}$ is obtained from $z_{n}$ by applying a function $G$ that is first-order definable in the structure on $D$ enriched by $\leq_{L}$. So this function is FA-recognizable. The pumping lemma now implies that the length of any linear combination of the $[u_{i},u_{k}]$, where $i<k<n$, is $O(n)$. This means that for large $n$ there are not enough strings of the allowed length to accommodate these $p^{\binom{n}{2}}$ linear combinations.

For the detail, we verify three claims. Note that the centre $Z=Z(F_{\infty})$ is a regular set. The first claim states that for each finite set $S\subseteq D$ there is $u\in D$ such that the map $ZS/Z\to Z$ given by $Zr\to[r,u]$ has range disjoint from $S$, and is injective.

Claim 3.2**.**

For each finite set $S\subseteq D$, there is a string $u\in D$ such that

• (i)

if $r\in S\setminus Z$ then $[r,u]\not\in S$;

• (ii)

for each $v,w\in S$ such that $[v,u]=[w,u]$, there is $c\in Z$ such that $c\circ v=w$.

Given $S$, we will write $u(S)$ for the $\leq_{L}$-least string $u$ satisfying the claim.

To see this, let $u=x_{k}$ where $k$ is so large that only $x_{i}$ with $i<k$ occur in the normal form of any element of $S$.

For (i) note that $r$ contains some $x_{i}$ with $i<k$, so the normal form of $[r,u]$ contains $[x_{i},x_{k}]$, while the normal form of an element of $S$ does not contain such commutators.

For (ii) let $v=\underline{\alpha}d$ with $d$ central. Then the normal form of $[v,u]$ ends in $\prod_{i<k}[x_{i},x_{k}]^{\alpha_{i}}$, which determines $\alpha$. This verifies the claim.

We now recursively define the sequences ${\left\langle{u_{n}}\right\rangle}_{n\in{\mathbb{N}}}$ and ${\left\langle{z_{n}}\right\rangle}_{n\in{\mathbb{N}}}$ in $D$. Let $z_{0}$ be the string representing the neutral element $1$. Suppose now that $z_{n}$ has been defined. Let

$u_{n}=u(V_{n})$ where $V_{n}=\{v\colon v\leq_{L}z_{n}\}$,

according to Claim 3.2. Next, let $z_{n+1}$ be the $\leq_{L}$-least string $z$ such that

[TABLE]

In the structure $(D,\circ,\leq_{L})$, one can define $u_{n}$ from $z_{n}$ in a uniform first order way. Hence, there is a function $G\colon D\to D$ that is first-order definable in this structure such that $z_{n+1}=G(z_{n})$ for each $n$. As mentioned in the introduction, its definability implies that its graph of $G$ can be recognized by a finite automaton. In particular, we have $|z_{n}|=O(n)$ since by the pumping lemma, the output of a function with regular graph is by at most a constant longer than the input.

In the following, we will write $u_{i,k}$ for $[u_{i},u_{k}]$ in case that $i\neq k$.

Claim 3.3**.**

$\langle u_{0},\ldots,u_{n}\rangle\subseteq V_{n+1}$* for each $n$.*

We use induction on $n$. For $n=0$ we have $\langle u_{0}\rangle\subseteq V_{1}$ because in (2) we can let $v,w$ be strings denoting the neutral element. For the inductive step, note that by the normal form (and freeness of $F_{\infty}$) each element of $\langle u_{0},\ldots,u_{n}\rangle$ has the form

$y=\prod_{i\leq n}u_{i}^{\alpha_{i}}\prod_{r<s\leq n}[u_{r,s}]^{\gamma_{r,s}}$

where $0\leq\alpha_{i},\gamma_{r,s}<p$. This can be rewritten as $wu_{n}^{\alpha_{n}}[v,u_{n}]$ where

$w=\prod_{i<n}u_{i}^{\alpha_{i}}\prod_{r<s<n}[u_{r,s}]^{\gamma_{r,s}}$ and $v=\prod_{k<n}u_{k}^{\gamma_{k,n}}$.

By inductive hypothesis $w,v\in V_{n}$. So the element $y$ is in $V_{n+1}$ by (2). This verifies the claim.

In the next claim we view elementary abelian $p$-groups as vector spaces over the field $GF(p)$.

Claim 3.4**.**

**

• (a)

The elements $Zu_{i}$ are linearly independent in $G/Z$.

• (b)

The elements $u_{i,k}$ are linearly independent in $Z$.

In both (a) and (b) we use induction over an upper bound $n$ on the indices. Both statements hold vacuously for $n=0$. For (a) note that $u_{n+1}^{i}\not\in V_{n+1}Z$ for each $i<p$: Otherwise there is $c\in Z$ such that $v:=u_{n+1}^{i}c\in V_{n+1}$. We have $u_{n+1}^{i}\not\in Z$ and hence $v\not\in Z$. Since $[v,u_{n+1}]=1$ and $1=z_{0}\in V_{n+1}$, this contradicts condition (i) of Claim 3.2 for $S=V_{n+1}$. Therefore, by the Claim 3.3, $Z\langle u_{n+1}\rangle\cap Z\langle u_{0},\ldots,u_{n}\rangle=0$.

For (b), inductively the $u_{i,k}$, $i<k\leq n$ form a basis for a subspace $T_{0}\subseteq Z$. The linear map $Zw\to[w,u_{n+1}]$ defined on $ZV_{n}/Z$ is injective by (ii) of Claim 3.2. So, by (a), the $[u_{i},u_{n+1}]$ for $i\leq n$ form a basis of a subspace $T_{1}\subseteq Z$. Then $T_{0}\cap T_{1}=\{0\}$: if $\sum_{i\leq n}{\alpha_{i}}[u_{i},u_{n+1}]=[\sum_{i\leq n}{\alpha_{i}}u_{i},u_{n+1}]\neq 0$ for some coefficients $\alpha_{i}$ with $0\leq\alpha_{i}<p$, then $r:=\sum_{i\leq n}{\alpha_{i}}u_{i}\in V_{n+1}\setminus Z$ by Claim 3.3, and hence $[r,u_{n+1}]\not\in V_{n+1}$ by condition (i) of Claim 3.2. By Claim 3.3 again this implies $[r,u_{n+1}]\not\in T_{0}$. This concludes the inductive step and verifies the claim.

Given $n$, by Claim 3.3 we have

$\prod_{i<k<n}[u_{i},u_{k}]^{\gamma_{i,k}}\in V_{n}$

for each array ${\left\langle{\gamma_{i,k}}\right\rangle}_{i<k<n}$ of exponents in $[0,p)$. By Claim 3.4 all these elements are distinct. Since $V_{n}$ consists of strings of length $O(n)$, we have $p^{n(n-1)/2}$ distinct strings of length $O(n)$, which is contradictory for large enough $n$. ∎

4. Quotients of $F_{\infty}$ with dependency between the commutators

As before, let $F_{\infty}$ denote the group in the variety $\mathcal{N}_{2,p}$ with free generators $x_{i}$, $i\in{\mathbb{N}}$. We next define groups $E,H$ in $\mathcal{N}_{2,p}$ as quotients of $F_{\infty}$. For $E$ we require that all the commutators $[x_{i},x_{\ell}]$, $i<\ell$, be equal. For $H$ we require that $[x_{i},x_{k}]=z_{k}$ for $i<k$ where the $z_{k}$ are linearly independent over ${\mathbb{Z}}/p{\mathbb{Z}}$. Formally, we define the groups via presentations in the variety $\mathcal{N}_{2,p}$:

[TABLE]

Note that $E$ is extra-special as discussed in the introduction. In contrast, the centre of $H$ has infinite dimension.

In a nilpotent group, each nontrivial normal subgroup intersects the centre non-trivially. This implies that every proper quotient of $E$ is abelian; in particular, $E$ is not residually finite. On the other hand, $H$ is residually a finite $p$-group, and hence is canonically embedded into its pro-$p$ completion. To see this, take an element $h\neq 1$ that can be written in terms of the generators $x_{0},\ldots,x_{n-1}$. Then $h\neq 1$ in the finite $p$-group which is the quotient of $H$ by the normal subgroup generated by the $x_{k}$, $k\geq n$ (which contains all the $z_{k}$, $k\geq n$).

We supply two algebraic facts supporting the claim that the FA presentations of the groups we provide in the next section will be nontrivial: neither can they be obtained from FA presentations of abelian groups, nor are they combinations of FA presentations of simpler components.

Proposition 4.1**.**

None of the groups $E,H$ has an abelian subgroup of finite index.

Proof.

It suffices to show this for $E$, because $H$ has $E$ as a quotient. So suppose $M$ is a subgroup of $E$ with finite index. There are $k<r<s$ such that $x_{r}x_{k}^{-1}\in M$ and $x_{s}x_{k}^{-1}\in M$. We have $[x_{r}x_{k}^{-1},x_{s}x_{k}^{-1}]=[x_{r},x_{s}][x_{r},x_{k}^{-1}][x_{k}^{-1},x_{s}]=z$, so these two elements of $M$ don’t commute. ∎

Proposition 4.2**.**

The groups $E$ and $H$ are indecomposable.

Proof.

$E$ is indecomposable because each of its proper quotients is abelian. Suppose that $H=A\times B$ for subgroups $A,B$. Write $Z=Z(H)$, $\widetilde{x}_{i}=x_{i}Z$, $\widetilde{A}=AZ/Z$, and $\widetilde{B}=BZ/Z$. For a string $\beta$ over ${\mathbb{Z}}$ we write $\widetilde{\beta}=\prod_{i<|\beta|}\widetilde{x}_{i}^{\beta_{i}}$. If $A\subseteq Z$ then all $\widetilde{x}_{i}$ are in $\widetilde{B}$, so $z_{k}=[x_{0},x_{k}]$ is in $B$ for each $k\geq 1$, and hence $B=H$. By symmetry we may assume that $A,B\not\subseteq Z$.

First suppose that $\widetilde{x}_{0}^{m}\in\widetilde{A}$ for some $m$ with $0<m<p$. Then there is a string $\beta$ of length at least $2$ such that $\widetilde{\beta}\in\widetilde{B}$. Hence $[x_{0}^{m},\underline{\beta}]=\prod_{0<i<|\beta|}z_{i}^{m\beta_{i}}\neq e$, contradiction.

Now assume otherwise. There are strings $\alpha,\beta$ over $\{0,\ldots,p-1\}$ with $\widetilde{\alpha}\in\widetilde{A},\widetilde{\beta}\in\widetilde{B}$ such that $\widetilde{x}_{0}=\widetilde{\alpha}+\widetilde{\beta}$. By assumption $\alpha_{k}\neq 0$ for some $k>0$, chosen to be least. Then $\beta_{0}=1-\alpha_{0}$, $\beta_{k}=-\alpha_{k}$ and $\beta_{r}=0$ for $0<r<k$. Hence, using that the commutator is bilinear, $[\underline{\alpha},\underline{\beta}]$ contains a factor $z_{k}^{\gamma}$ where $\gamma=\alpha_{k}\beta_{0}-\alpha_{0}\beta_{k}=\alpha_{k}$. Thus $[\underline{\alpha},\underline{\beta}]\neq e$, contradiction. ∎

5. The groups $E$ and $H$ are word automatic

Recall that, fixing a prime $p$, the groups $E$ and $H$ are defined via (3) and (4). We will describe FA presentations of these groups based on the alphabet

$\Sigma=\{0,\ldots,p-1\}$.

The variables $\alpha,\beta,\gamma$ will denote strings over $\Sigma$. Recall that we write $\underline{\alpha}$ for $\prod_{i}x_{i}^{\alpha_{i}}$, where the $x_{i}$ are the generators given by the presentations. Using the normal form for elements in $F_{\infty}$ given in (1), each element of $E$ can be written in the form $z^{v}\cdot\underline{\alpha}$ where $0\leq v<p$.

In this section all arithmetic is modulo $p$. Strings $\alpha,\beta$ in expressions such as $\alpha+\beta$ are thought of as extended by [math]s if necessary, and will be added component-wise.

In the case of the group $E$, one notes that for $n=\max(|\alpha|,|\beta|)$,

[TABLE]

This is because one can calculate $\underline{\alpha}\cdot\underline{\beta}=\prod_{k}x_{k}^{\alpha_{k}}\prod_{i}x_{i}^{\beta_{i}}$ by, for decreasing positive $k$, moving terms $x_{k}^{\alpha_{k}}$ to the right past the terms $x_{i}^{\beta_{i}}$ for $i=0,\ldots,k-1$, and then joining it with $x_{k}^{\beta_{k}}$ to form $x_{k}^{\alpha_{k}+\beta_{k}}$. Each such move creates a factor $z^{-\alpha_{k}\beta_{i}}$ in the centre of $E$.

Proposition 5.1**.**

The group $E$ is word automatic.

Proof.

An element $z^{v}\cdot\underline{\alpha}$ is represented by the string $v\alpha$. The domain consists of the strings $v\alpha$ such that $\alpha$ is empty, or its last entry is not [math].

We describe an FA that checks (5), and hence correctly verifies the binary group operation. It processes an input composed of a triple of strings $v\alpha,w\beta,r\gamma$ stacked on three tracks. If necessary, we extend the strings by [math]s to make them have length $1+n$ where $n=\max(|\alpha|,|\beta|,|\gamma|)$. So the FA processes an input over the alphabet $\Sigma^{3}$ in this format:

s_{-1}s_{0}\ldots s_{n-1}=\begin{array}[]{|l|l|l|l|l|l|l}\hline\cr v&\alpha_{0}&\alpha_{1}&\cdots&\alpha_{n-1}\\ \hline\cr w&\beta_{0}&\beta_{1}&\cdots&\beta{n-1}\\ \hline\cr r&\gamma_{0}&\gamma_{1}&\cdots&\gamma_{n-1}\\ \hline\cr\end{array}

When the FA scans the first stack symbol $s_{-1}$, it stores it in the constant size memory given by the state. As it scans $s_{k}$ for $k=0,\ldots,n-1$ it checks that $\alpha_{k}+\beta_{k}=\gamma_{k}$ (if this fails, it enters a rejecting state and remains in it until the whole input has been scanned). The FA stores the current value $\sum_{i<k}\beta_{i}$ in its constant size internal memory. It adds $\alpha_{k}(\sum_{i<k}\beta_{i})$ to a variable $x$ ranging over $\Sigma$, with initial value [math], also thought of as stored in the internal memory. After scanning the last symbol it checks whether $v+w-x=r\mod p$, and accepts accordingly. ∎

Each element of $H$ can be written in the form

$\prod_{s<|\alpha|}z_{s}^{v_{s}}\cdot\underline{\alpha}$

where $0\leq v_{s}<p$. The central elements are the ones where $\alpha$ consists only of [math]s. If $n=\max(|\alpha|,|\beta|)$, similar to (5) we have in $H$ that

[TABLE]

Proposition 5.2**.**

The group $H$ is word automatic.

Proof.

An element $h$ of the group $H$ will be represented by a pair of strings $\alpha,v$ over $\Sigma$, of the same length $n$, such that if $n>0$ then $v_{0}=0$, not both end in [math], and $h=\prod_{k=1}^{n}z_{k}^{v_{k}}\cdot\underline{\alpha}$. These strings are written on two tracks, with $\alpha$ on top of $v$.

We describe an FA that checks (6), and thus correctly verifies the group operation. It processes strings over the alphabet $\Sigma^{6}$ of the format

s_{0}\ldots s_{n-1}=\begin{array}[]{|l|l|l|l|l|l|l}\hline\cr\alpha_{0}&\alpha_{1}&\cdots&\alpha_{n-1}\\ \hdashline v_{0}&v_{1}&\cdots&v_{n-1}\\ \hline\cr\beta_{0}&\beta_{1}&\cdots&\beta{n-1}\\ \hdashline w_{0}&w_{1}&\cdots&w_{n-1}\\ \hline\cr\gamma_{0}&\gamma_{1}&\cdots&\gamma_{n-1}\\ \hdashline r_{0}&r_{1}&\cdots&r_{n-1}\\ \hline\cr\end{array}.

At the beginning, the FA scans $s_{0}$ and checks that $v_{0}=w_{0}=r_{0}=0$. As in the case of $E$, when the FA scans $s_{k}$ for $k\geq 0$, it checks that $\alpha_{k}+\beta_{k}=\gamma_{k}$, and stores the current value $\sum_{i<k}\beta_{i}$ in its constant size internal memory. However, now it also checks whether

$v_{k}+w_{k}-\alpha_{k}(\sum_{i<k}\beta_{i})=r_{k}\mod p$

(which holds trivially if $k=0$). If any of these checks fail it enters a rejecting state. Otherwise, when the whole input is scanned it accepts.∎

Büchi automata are (nondeterministic) finite automata that work on infinite words. Such a word is accepted if some computation processing it is infinitely often in an accepting state. Büchi automatic structures were first considered by Hodgson [7], who called them macro-automatic. For background see [13, Section 2.1]. We sketch an example of a non-abelian uncountable group that is Büchi automatic in a nontrivial way.

Fact 5.3**.**

The pro-$p$ completion $\hat{H}_{p}$ of $H_{p}$ is Büchi automatic.

Proof.

For each pair of infinite words $\alpha$, $v$ over $\{0,\ldots,p-1\}$, in $\hat{H}_{p}$ one can form the limits $\prod_{i}x_{i}^{\alpha_{i}}\colon=\lim_{n}\prod_{i<n}x_{i}^{\alpha_{i}}$ and $\prod_{k>0}z_{k}^{v_{k}}$. An element $h$ of the group $\hat{H}_{p}$ is represented by a pair of infinite words $\alpha,v$ such that $h=\prod_{k>0}z_{k}^{v_{k-1}}\cdot\prod_{i}x_{i}^{\alpha_{i}}$. We can use the same automaton as above, now working on infinite words, to verify the binary group operation on $\hat{H}_{p}$. ∎

6. Constructing word automatic groups via cocycles

We review some well-known facts on central extensions of abelian groups. Given two abelian groups $A$ and $Q$, a central extension of $Q$ by $A$ is an exact sequence $0\to A\to L\to Q\to 0$ such that $A\subseteq Z(L)$. Cocycles are used to describe such extensions. A cocycle is a function $f\colon Q\times Q\to A$ such that

[TABLE]

On $Q\times A$, the operation

[TABLE]

defines a group $L_{f}$, which is abelian iff $f$ is symmetric. It is easy to verify that with the maps $a\to(0,a)$ and $(u,a)\to u$, one obtains an exact sequence $0\to A\to L_{f}\to Q\to 0$. The inverse of $(u,a)$ is $(-u.-a-f(u,-u))$. For associativity, if we calculate $[(u,a)+(v.b)]+(w,c)$, the “correcting term” in the second component on the right side is $f(u,v)+f(u+v,w)$. If we calculate $(u,a)+[(v.b)+(w,c)]$, the correcting term is $f(v,w)+f(u,v+w)$.

Conversely, given an exact sequence $0\rightarrow A\rightarrow L\rightarrow Q\rightarrow 0$ with $A\leq Z(L)$, to determine a 2-cocycle $c\colon Q\times Q\rightarrow A$ that yields an equivalent extension, one picks a set $T$ of coset representatives for $A$ in $L$. Fixing a bijection $Q\to T$ and writing $\overline{q}$ for the image of $q$, the cocycle is given by

$f(q_{0},q_{1})\colon=\overline{q_{0}\phantom{+}}\cdot\overline{q_{1}\phantom{+}}\cdot(\overline{q_{0}+q_{1}})^{-1}\in A$.

For detailed background see e.g.  Fuchs [5, Ch. 9] (who calls these objects “extensions of $A$ by $Q$”, but uses the same order, first $Q$ then $A$, in the notation).

Proposition 6.1**.**

Let $A$ and $Q$ be word automatic abelian groups. Let $L$ be a central extension of $Q$ by $A$, given by an exact sequence $0\rightarrow A\rightarrow L\rightarrow Q\rightarrow 0$ with $A\leq Z(L)$. Suppose some cocycle $f\colon Q\times Q\rightarrow A$ describing this extension is FA recognizable. Then $L$ is word automatic.

Proof.

$L$ can be constructed as the set $Q\times A$ with the operation given above. So $L$ can be interpreted in a first-order way in the word automatic two-sorted structure $(Q\sqcup A,+_{Q},+_{A},f)$. This shows that $L$ is word automatic. ∎

Thus, if we can choose $T$ so that the corresponding cocycle $f$ can be computed by a finite automaton, we obtain an FA-presentation for $L$. As an example of how to apply Prop. 6.1, we revisit a group, introduced in [15, Example 12], that does not have an abelian subgroup of finite index. We give a short proof based on cocycles that this group is word automatic.

Example 6.2**.**

Let the group $L$ have generators $x,y_{i},z_{k}$ ($i,k\in{\mathbb{N}}$) subject to the relations

$y_{i}^{2}=z_{k}^{2}=1\qquad[y_{i},z_{k}]=[y_{i},x]=1\qquad z_{i}^{-1}xz_{i}=xy_{i}$

We have an exact sequence $0\rightarrow A\rightarrow L\rightarrow Q\rightarrow 0$ where $A=Z(L)=\langle x^{2},\{y_{i}\}_{i\in{\mathbb{N}}}\rangle$, and $Q:=L/A$ is the direct sum of groups $Ax$ and $Az_{i}$. Write $v=Ax$ and $w_{i}=Az_{i}$, so that $v^{2}=w_{i}^{2}=1$ in $Q$. Elements of $Q$ have a normal form $q_{s,\alpha}=v^{s}\cdot\prod_{i}w_{i}^{\alpha_{i}}$ where $s=0,1$ and $\alpha$ is a bit string (thought to be extended by [math]s if necessary).

The transversal $T$ consists of the elements $\overline{q_{s,\alpha}}=x^{s}\prod_{i}z_{i}^{\alpha_{i}}\in L$. Using that $z_{i}x=xz_{i}y_{i}$, one verifies that the corresponding 2-cocycle is

$c(q_{s,\alpha},q_{t,\beta})=\overline{q_{s,\alpha}\phantom{+}}\cdot\overline{q_{t,\beta}\phantom{+}}\cdot(\overline{q_{s,\alpha}+q_{t,\beta}}{\,})^{-1}=x^{2s+t}\prod_{i}y_{i}^{\gamma_{i}}$

where $\gamma_{i}=(s+2t)\alpha_{i}+(s+t)\beta_{i}\mod 2$. It is clear that this cocycle can be computed by an FA.

Remark 6.3**.**

Nies and Semukhin [14, Thm. 4.2] showed that an abelian group that has a word automatic normal subgroup of finite index of is in itself word automatic. By Prop 6.1 this actually holds without the restriction to abelian groups.**

Remark 6.4**.**

In Section 5 we showed that the groups $E_{p}$ and $H_{p}$ are word automatic. The automata in the proofs can also be somewhat simplified using cocycles. The elements of the form $\underline{\alpha}$ form a transversal for the extension. The cocycles are then given in (5) and (6), respectively. For $H$, say, the cocycle maps pairs $\widetilde{\alpha},\widetilde{\beta}$ of elements of $Q=H/Z(H)$, where $|\alpha|=|\beta|=n$, to elements $\prod_{k=1}^{n-1}z_{k}^{r_{k}}\in Z(H)$. So the automaton verifying the cocycle processes strings of the format

t_{0}\ldots t_{n-1}=\begin{array}[]{|l|l|l|l|l|l|l}\hline\cr\alpha_{0}&\alpha_{1}&\cdots&\alpha_{n-1}\\ \hline\cr\beta_{0}&\beta_{1}&\cdots&\beta{n-1}\\ \hline\cr 0&r_{1}&\cdots&r_{n-1}\\ \hline\cr\end{array}.

As it scans the symbols $t_{0},\ldots,t_{k},\ldots$, it stores the current value $\sum_{i<k}\beta_{i}$ in its constant size internal memory. It checks whether $-\alpha_{k}(\sum_{i<k}\beta_{i})=r_{k}\mod p$; else it enters a rejecting state. Otherwise, when the whole input has been scanned, it accepts. **