1. Introduction

All groups considered will be finitely generated and residually finite. Two groups are said to be profinitely isomorphic if they share the same set of finite quotients, and a group $G$ is profinitely rigid (within a class $\mathcal {C}$ ) if any group (within $\mathcal {C}$ ), which is profinitely isomorphic to $G$ is isomorphic to $G$ . One can ask which groups within which classes are profinitely rigid, though this question can be subtle even when the class is very restricted: finitely generated abelian groups are profinitely rigid, free groups are profinitely rigid among themselves but it is an open question due to Remeslennikov [Reference Khukhro and Mazurov11], Question 5.48] whether this holds among finitely generated groups, and there are pairs of non-isomorphic virtually free (even virtually $\mathbb {Z}$ !) groups which are profinitely isomorphic [Reference Baumslag1].

Grothendieck asked if there are groups that are profinitely isomorphic to one of their proper, non-isomorphic subgroups: Platonov and Tavgen [Reference Platonov and Tavgen12] provide examples of this phenomenon for finitely generated groups, and Bridson and Grunewald [Reference Bridson and Grunewald2], Theorem 1.1] for finitely presented groups. For a discussion of these facts, as well as a broader introduction to profinite rigidity of groups, see for instance [Reference Reid13].

Herein, we consider profinite rigidity within the class of free-by-cyclic groups. A free-by-cyclic group is a semidirect product $G:=F_r \rtimes _\varphi \mathbb {Z}$ , where $F_r$ denotes the free group of finite rank $r$ . If two automorphisms $\varphi, \psi$ of $F_r$ represent conjugate or conjugate inverse elements of $\textrm {Out}(F_r)$ then they define isomorphic free-by-cyclic groups [Reference Bogopolski, Martino and Ventura4], Lemma 2.1]. The converse holds when the abelianisation has rank 1, $b_1(G)=1$ [Reference Bogopolski, Martino and Ventura4], Theorem 2.4], though not in general [Reference Bogopolski, Martino and Ventura4], p1678].

We provide the first known example of a hyperbolic free-by-cyclic group that is profinitely rigid amongst free-by-cyclic groups.

Hughes and Kudlinska [Reference Hughes and Kudlinska10] have shown that if $G$ has $b_1(G)=1$ then in many cases $G$ is almost profinitely rigid: it shares a profinite isomorphism class with at most finitely many non-isomorphic free-by-cyclic groups. These cases include when the defining automorphism is irreducible (it has no non-trivial preserved free factor system) and when the rank of the kernel is 3. Building off work of Bridson, Reid and Wilton [Reference Bridson, Reid and Wilton8] they also show that if the kernel has rank 2 then $G$ is profinitely rigid among free-by-cyclic groups. Note that in the rank 2 case $G$ is never hyperbolic.

In a different direction Bridson and Piwek have very recently shown that free-by-cyclic groups with centre [Reference Bridson and Piwek5], Theorem 1.1], or where the cyclic group is instead required to be finite cyclic [Reference Bridson and Piwek5], Theorem 1.2], are profinitely rigid among respectively all free-by-(infinite cyclic) groups and free-by-(finite cyclic) groups.

Following the analogy between free-by-cyclic groups and three manifolds, we recall two cognate results on the profinite rigidity of three manifolds. First, the fundamental group of the figure-eight knot complement is profinitely rigid amongst all fundamental groups of three manifolds [Reference Bridson and Reid6], Theorem A]: the proof goes via the identification of this manifold with a once punctured torus bundle over the circle, which algebraically corresponds to some $F_2 \rtimes \mathbb {Z}$ . Second, and stronger, the fundamental group of the Weeks manifold (the unique closed orientable hyperbolic three manifold of minimal volume) is profinitely rigid, with no need to restrict the class concerned [Reference Bridson, McReynolds, Reid and Spitler3], Theorem 9.1].

Proof of the theorem. The proof will be an application of Hughes and Kudlinska’s characterisation of properties detected by the profinite isomorphism class of a free-by-cyclic group (Theorem2), together with Hillen’s work (Corollary3) controlling stretch factors of elements of $\textrm {Out}(F_r)$ :

We briefly recap the concepts introduced within the theorem. Brinkmann showed that $G$ is hyperbolic if and only if its defining automorphism is atoroidal: it has no periodic conjugacy classes [Reference Brinkmann7], Theorem 1.2]. The stretch factor $\lambda$ of an automorphism is defined to be

\begin{equation*} \sup _{w \in F_r} \limsup \sqrt [n]{\| \varphi ^n(w) \|},\end{equation*}

where $\|w\|$ is the cyclically reduced word length of the element $w$ ; it records “how fast” elements grow under repeated application of the automorphism, and is an $\textrm {Out}(F_r)$ -conjugacy class invariant. Elements of $\textrm {Out}(F_r)$ can and often do have different stretch factors from their inverses, so we must record both. Note that the set $\{\lambda _H^+, \lambda _H^-\}$ is well defined since the profinite isomorphism class of a group determines its abelianisation (see for instance [Reference Reid13], Remark 3.2]), and in particular $H$ must also have $b_1(H)=1$ and so there is a unique (up to inverting and $\textrm {Out}(F_r)$ -conjugacy) defining automorphism for $H$ .

Proof of Theorem 1. First we check that $G$ satisfies the hypotheses of Theorem2. A quick computation with the abelianisation verifies that $b_1(G)=1$ , while hyperbolicity follows from Brinkmann’s theorem [Reference Brinkmann7], Theorem 1.2] since the single fold representative has no periodic Nielsen paths so there cannot be a periodic conjugacy class.

Now, suppose $H$ is another free-by-cyclic group that is profinitely isomorphic to $G$ . From Theorem2 we know that $H$ is some $F_3 \rtimes \mathbb {Z}$ , that it is hyperbolic, and that the defining automorphism and its inverse will have the same stretch factor(s) as $\psi$ and its inverse; in particular one of them must be $\lambda$ . Let $\varphi$ be the choice with smaller stretch factor.

It follows from Brinkmann’s theorem that $\varphi$ is atoroidal. We also observe, as for instance in the proof of [Reference Hughes and Kudlinska10], Corollary F], that if an element of $\textrm {Out}(F_3)$ is atoroidal then it must be irreducible (the point is that a preserved free factor would have rank 1 or 2, and in either case there is a periodic conjugacy class). But then, by Corollary3, we have that $\varphi$ and $\psi$ are conjugate as elements of $\textrm {Out}(F_3)$ , and so they define isomorphic free-by-cyclic groups.