1. Introduction
A subgroup 
$H \leqslant G$ is called separable if for all 
$g \in G \setminus H$ there exists a finite quotient 
$q \colon G \to Q$ such that 
$q(g) \notin q(H)$. A question of Reid [Reference Reid26, Question 3.5] asks whether all convex-cocompact subgroups of mapping class groups (as defined in [Reference Farb and Mosher9], see also [Reference Kent IV and Leininger19] for a characterization) are separable. This was first verified for virtually cyclic subgroups [Reference Leininger and McReynolds21], and the full conjecture is known conditionally on the residual finiteness of hyperbolic groups [Reference Behrstock, Hagen, Martin and Sisto3]. Hagen and Sisto show in [Reference Hagen and Sisto17] that certain examples of free convex-cocompact subgroups, constructed by Mj [Reference Mj22], are separable. In order to do so, they prove the following criterion.
Theorem ([Reference Hagen and Sisto17, Theorem 1.1])
Let 
$g \geqslant 2$, and let 
$H \leqslant \operatorname{MCG}(\Sigma_g)$. Suppose that H is torsion-free, malnormal, and convex-cocompact. If the preimage of H under the natural quotient map 
$\operatorname{MCG}(\Sigma_g \setminus \{p\}) \to \operatorname{MCG}(\Sigma_g)$, for some point 
$p \in \Sigma_g$, is conjugacy separable, then H is separable.
Remark.
The theorem is stated for 
$g \geqslant 1$; however, in that case, 
$\operatorname{MCG}(\Sigma_g)$ is virtually free so every finitely generated subgroup is separable. We focus on the case 
$g \geqslant 2$ for coherence with the results of this note.
A group G is said to be conjugacy separable if for all non-conjugate elements 
$g, h \in G$ there exists a finite quotient 
$q \colon G \to Q$ such that q(g) and q(h) are non-conjugate. Our first result removes the other hypotheses on H, answering [Reference Hagen and Sisto17, Question 1.4].
Theorem A.
Let 
$g \geqslant 2$, and let 
$H \leqslant \operatorname{MCG}(\Sigma_g)$. If the preimage of H under the natural quotient map 
$\operatorname{MCG}(\Sigma_g \setminus \{p\}) \to \operatorname{MCG}(\Sigma_g)$, for some point 
$p \in \Sigma_g$, is conjugacy separable, then H is separable.
We can replace the hypothesis of conjugacy separability by a more geometric one, which goes in the direction of Reid’s question [Reference Reid26, Question 3.5]. This is the same way that Hagen and Sisto verify that their criterion holds for the groups constructed by Mj.
Corollary B.
Let 
$g \geqslant 2$, and let 
$H \leqslant \operatorname{MCG}(\Sigma_g)$ be a convex-cocompact subgroup. If the preimage of H under the natural quotient map 
$\operatorname{MCG}(\Sigma_g \setminus \{p\}) \to \operatorname{MCG}(\Sigma_g)$, for some point 
$p \in \Sigma_g$, acts properly and cocompactly on a 
$\operatorname{CAT(0)}$ cube complex, then H is separable.
Notice that Corollary B applies to all groups constructed in [Reference Mj22], without needing to modify the construction to ensure malnormality, as in [Reference Hagen and Sisto17, Section 5].
Theorem A will follow from a more general result.
Theorem C.
Let G be a finitely generated group with trivial centre, and let 
$H \leqslant \operatorname{Out}(G)$. Suppose that 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic and has no non-trivial finite normal subgroups. If the preimage of H under the natural quotient map 
$\operatorname{Aut}(G) \to \operatorname{Out}(G)$ is conjugacy separable, then H is separable.
Thanks to recent works proving acylindrical hyperbolicity of automorphism groups [Reference Escalier and Horbez8, Reference Genevois10–Reference Genevois and Martin13], this can be applied in several contexts. We isolate two instances.
Corollary D.
Let G be a torsion-free hyperbolic group, and let 
$H \leqslant \operatorname{Out}(G)$. If the preimage of H under the natural quotient map 
$\operatorname{Aut}(G) \to \operatorname{Out}(G)$ is conjugacy separable, then H is separable.
Corollary E.
Let Γ be a finite simplicial graph that does not decompose as a join of two non-empty subgraphs. Let 
$A_\Gamma$ be the corresponding right-angled Artin group, and let 
$H \leqslant \operatorname{Out}(A_\Gamma)$. If the preimage of H under the natural quotient map 
$\operatorname{Aut}(A_\Gamma) \to \operatorname{Out}(A_\Gamma)$ is conjugacy separable, then H is separable.
Both of these corollaries apply to 
$\operatorname{Out}(F_n)$. In this case, there are notions of convex-cocompact subgroups [Reference Dowdall and Taylor6, Reference Dowdall and Taylor7, Reference Hamenstädt and Hensel16]; the corresponding preimage in 
$\operatorname{Aut}(F_n)$ is hyperbolic [Reference Dowdall and Taylor7], but we do not know of instances in which conjugacy separability of such groups is known (besides the case of virtually cyclic subgroups).
Question F.
Are there examples of subgroups 
$H \leqslant \operatorname{Out}(F_n)$ that are separable, not virtually cyclic, and satisfy some version of convex-cocompactness?
We remark that free-by-cyclic groups 
$F_n \rtimes \mathbb{Z}$ often have non-separable subgroups [Reference Kudlinska20].
The proof of Theorem C involves an element of G that recognises non-inner automorphisms. The existence of such an element relies on the acylindrical hyperbolicity of 
$\operatorname{Aut}(G)$, and is what allows to strengthen and generalise the criterion of Hagen and Sisto, with a shorter proof. It partially answers [Reference Hagen and Sisto17, Questions 1.5] (Proposition 2.6) and [Reference Hagen and Sisto17, Question 1.6] (Proposition 2.8).
2. Proofs
We will use π to denote the quotient map 
$\pi \colon \operatorname{Aut}(G) \to \operatorname{Out}(G)$, and for a subgroup 
$H \leqslant \operatorname{Out}(G)$ we denote 
$\widetilde{H} {:=} \pi^{-1} H \leqslant \operatorname{Aut}(G)$. The starting point is the following criterion for separability, which in turn is based on Grossman’s criterion for residual finiteness [Reference Grossman14].
Proposition 2.1. ([Reference Hagen and Sisto17, Proposition 2.5])
Let G be a finitely generated group with trivial centre. Let 
$H \leqslant \operatorname{Out}(G)$ and 
$\alpha \in \operatorname{Aut}(G)$. Suppose that:
1. There exists
$x \in G$ such that
$\alpha(x) \neq h(x)$ for all
$h \in \widetilde{H}$;2.
$\widetilde{H}$ is conjugacy separable.
Then there exists a finite quotient 
$q \colon \operatorname{Out}(G) \to Q$ such that 
$q(\pi(\alpha)) \notin q(H)$.
Let us formulate the special case that we will use:
Corollary 2.2. Let G be a finitely generated group with trivial centre and let 
$H \leqslant \operatorname{Out}(G)$. Suppose that:
1. There exists
$\gamma \in \operatorname{Inn}(G)$ such that the centraliser of γ in
$\operatorname{Aut}(G)$ is
$\langle \gamma \rangle$;2.
$\widetilde{H}$ is conjugacy separable.
Then H is separable.
Proof. We show that Proposition 2.1 holds for H and an arbitrary 
$\alpha \in \operatorname{Aut}(G) \setminus \widetilde{H}$. Choose 
$x \in G$ such that the corresponding inner automorphism γx is as in the first assumption of the corollary. Suppose that 
$\alpha(x) = h(x)$ for some 
$h \in \widetilde{H}$. Then 
$h^{-1} \alpha (x) = x$, and so 
$h^{-1} \alpha$ belongs to the centraliser of γx in 
$\operatorname{Aut}(G)$. By the choice of x, we have 
$h^{-1} \alpha \in \langle \gamma_x \rangle \leqslant \operatorname{Inn}(G) \leqslant \widetilde{H}$, which contradicts 
$\alpha \notin \widetilde{H}$.
Acylindrical hyperbolicity of 
$\operatorname{Aut}(G)$ ensures that the first item holds.
Lemma 2.3. Let G be a group such that 
$\operatorname{Inn}(G)$ is infinite and 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic and has no non-trivial finite normal subgroups. Then there exists 
$\gamma \in \operatorname{Inn}(G)$ such that the centraliser of γ in 
$\operatorname{Aut}(G)$ is 
$\langle \gamma \rangle$.
In fact, such a γ can be found by performing a simple random walk on G: see Proposition 2.8 and its proof.
Proof. Recall that a subgroup of an acylindrically hyperbolic group is called suitable if it is non-elementary and does not normalise any non-trivial finite normal subgroup. In an acylindrically hyperbolic group with no non-trivial finite normal subgroups, every infinite normal subgroup is suitable [Reference Osin25, Lemma 2.4]. In particular, 
$\operatorname{Inn}(G) \leqslant \operatorname{Aut}(G)$ is suitable. Therefore, there exists an inner automorphism 
$\gamma \in \operatorname{Inn}(G)$ such that the elementary closure of γ in 
$\operatorname{Aut}(G)$ is reduced to 
$\langle \gamma \rangle$ [Reference Hull18, Lemma 5.6]. In particular, the centraliser of γ in 
$\operatorname{Aut}(G)$ is reduced to 
$\langle \gamma \rangle$ [Reference Dahmani, Guirardel and Osin4, Corollary 6.6].
For the next applications, we will use the following criterion to check that an automorphism group has no non-trivial finite normal subgroups. We say that G has the unique root property if 
$x^n = y^n$ for some 
$x, y \in G, n \geqslant 1$ implies x = y.
Lemma 2.4. Let G be a group with trivial centre and with the unique root property. Then 
$\operatorname{Aut}(G)$ has no non-trivial finite normal subgroups.
Proof. Suppose that 
$N \leqslant \operatorname{Aut}(G)$ is a finite normal subgroup. The action of 
$G \cong \operatorname{Inn}(G)$ on N by conjugacy has a finite index kernel K. Then every element of K commutes with every element of N, in other words, automorphisms in N fix K pointwise. Now let 
$\alpha \in N$ and 
$x \in G$. Let 
$n \geqslant 1$ be such that 
$x^n \in K$, so xn is fixed by α. Then 
$x^n = \alpha(x^n) = \alpha(x)^n$, so by the unique root property 
$\alpha(x) = x$. This shows that α fixes every element of G and we conclude.
Proof of Corollary D
If G is torsion-free elementary hyperbolic, then G is either trivial or isomorphic to 
$\mathbb{Z}$, and in both cases 
$\operatorname{Out}(G)$ is finite, so all subgroups are separable.
If G is torsion-free non-elementary hyperbolic, then 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic [Reference Genevois and Horbez12, Theorem 1.3]. Moreover, G has trivial centre and the unique root property [Reference Bartholdi and Bogopolski2, Lemma 2.2] and so 
$\operatorname{Aut}(G)$ has no non-trivial finite normal subgroups by Lemma 2.4. Therefore, Theorem C applies.
Proof of Corollary E
Let 
$G = A_\Gamma$ be as in the statement. If Γ has at most one vertex, then G is either trivial or isomorphic to 
$\mathbb{Z}$, and in both cases 
$\operatorname{Out}(G)$ is finite, so all subgroups are separable.
If Γ has at least two vertices, then 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic [Reference Genevois11, Theorem 1.5]. Moreover, G has trivial centre and the unique root property [Reference Duchamp and Krob5, 3-2) and 3-3)] and so 
$\operatorname{Aut}(G)$ has no non-trivial finite normal subgroups by Lemma 2.4. Therefore, Theorem C applies.
For the results on mapping class groups, we apply Corollary D to the special case of surface groups.
Proof of Theorem A
Recall the Dehn–Nielsen–Baer Theorem: 
$\operatorname{MCG}(\Sigma_g)$ can be identified with an index-2 subgroup of 
$\operatorname{Out}(\pi_1(\Sigma_g))$, and 
$\operatorname{MCG}(\Sigma_g \setminus \{p \})$ is the corresponding index-2 subgroup of 
$\operatorname{Aut}(\pi_1(\Sigma_g))$. Under these identifications, given a subgroup 
$H \leqslant \operatorname{MCG}(\Sigma_g)$, its preimage in 
$\operatorname{MCG}(\Sigma_g \setminus \{p \})$ is the same as its preimage in 
$\operatorname{Aut}(\pi_1(\Sigma_g))$. Since 
$\pi_1(\Sigma_g)$ is torsion-free non-elementary hyperbolic, we can apply Corollary D to get separability of H in 
$\operatorname{Out}(\pi_1(\Sigma_g))$, which then implies separability in 
$\operatorname{MCG}(\Sigma_g)$.
Proof of Corollary B
By Theorem A, it suffices to show that, under the hypotheses, the preimage 
$\widetilde{H}$ is conjugacy separable. By convex-cocompactness of H, 
$\widetilde{H}$ is hyperbolic [Reference Farb and Mosher9, Reference Hamenstädt15]. By assumption, 
$\widetilde{H}$ acts properly and cocompactly on a 
$\operatorname{CAT(0)}$ cube complex, and so it is virtually compact special [Reference Agol1, Reference Wise27]. It follows that 
$\widetilde{H}$ is conjugacy separable [Reference Minasyan and Zalesskii24].
Let us end by addressing two questions from [Reference Hagen and Sisto17], which ask for elements that recognise non-inner automorphisms. The observation behind Lemma 2.3 allows to answer both, under the assumptions of Theorem C. The two questions are asked for torsion-free acylindrically hyperbolic groups, with the case of hyperbolic groups being singled out. It is an open question whether the automorphism group of a finitely generated acylindrically hyperbolic group is always acylindrically hyperbolic [Reference Genevois10, Question 1.1].
Question 2.5. ([Reference Hagen and Sisto17, Question 1.5])
Let G be a torsion-free acylindrically hyperbolic group, and let 
$\phi_1, \ldots, \phi_n$ be non-inner automorphisms of G. Does there exist 
$x \in G$ with x and 
$\phi_i(x)$ non-conjugate for all i?
Proposition 2.6. Let G be a group such that 
$\operatorname{Inn}(G)$ is infinite and 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic and has no non-trivial finite normal subgroups: for instance, a torsion-free non-elementary hyperbolic group. Then there exists 
$x \in G$ with the following property: for every non-inner automorphism ϕ, x and 
$\phi(x)$ are non-conjugate.
Recall that torsion-free non-elementary hyperbolic groups do indeed satisfy the hypotheses, as we saw in the proof of Corollary D.
Proof. Let x be such that γx satisfies the statement of Lemma 2.3. Suppose that 
$\phi(x) = hxh^{-1}$. Then 
$\gamma_h^{-1} \phi$ fixes x, so it centralises γx. By the choice of x, we have 
$\gamma_h^{-1} \phi \in \langle \gamma_x \rangle \leqslant \operatorname{Inn}(G)$ and so 
$\phi \in \operatorname{Inn}(G)$.
Question 2.7. ([Reference Hagen and Sisto17, Question 1.6])
Let G be a torsion-free acylindrically hyperbolic group, let ϕ be a non-inner automorphism of G, and let 
$(w_n)$ be a simple random walk on G. Is it true that, with probability going to 1 as n goes to infinity, wn is not conjugate to 
$\phi(w_n)$?
Note that finite generation is implicit in this question, as simple random walks are not defined over infinitely generated groups.
Proposition 2.8. Let G be a finitely generated group with trivial centre such that 
$\operatorname{Aut}(G)$ is acylindrically hyperbolic and has no non-trivial finite normal subgroups: for instance, a torsion-free non-elementary hyperbolic group. Let 
$(w_n)$ be a simple random walk on G. Then, with probability going to 1 as n goes to infinity, wn has the following property: for every non-inner automorphism ϕ, wn and 
$\phi(w_n)$ are non-conjugate.
Proof. We will use a result from [Reference Maher and Sisto23], from which we recall some terminology, in the special case we are interested in. We call an element 
$\alpha \in \operatorname{Aut}(G)$ asymmetric if its elementary closure is 
$\langle \alpha \rangle$. Let µ be a probability distribution on 
$\operatorname{Aut}(G)$. We say that µ is admissible (with respect to a fixed acylindrical action) if the support of µ is bounded and generates a non-elementary subgroup containing an asymmetric element. Let ν be the uniform measure on a finite generating set of G, and let µ be the pushforward of ν under the map 
$G \to \operatorname{Aut}(G)$. The simple random walk 
$(w_n)$ is generated by ν, and it induces a random walk 
$(\gamma_{w_n})$ generated by µ. Since the support of µ is finite, and it generates 
$\operatorname{Inn}(G)$ which is non-elementary and contains an asymmetric element (Lemma 2.3), µ is admissible. The cyclic subgroups 
$\langle \gamma_{w_n} \rangle$ are called random subgroups of 
$\operatorname{Aut}(G)$ (for k = 1) in the language of [Reference Maher and Sisto23].
Now we can apply [Reference Maher and Sisto23, Theorem 2.5], which states that with probability going to 1 as n goes to infinity, 
$\gamma_{w_n}$ is an asymmetric element of 
$\operatorname{Aut}(G)$. This implies that the centraliser of 
$\gamma_{w_n}$ is 
$\langle \gamma_{w_n} \rangle$ [Reference Dahmani, Guirardel and Osin4, Corollary 6.6], and we conclude as in Proposition 2.6.
Acknowledgements
The author thanks Mark Hagen, Jonathan Fruchter, Monika Kudlinska, Alessandro Sisto, Ric Wade and Henry Wilton for useful comments on a first version; and the anonymous referee for more useful comments, especially for suggesting Proposition 2.8. The author is supported by the Herchel Smith Postdoctoral Fellowship Fund.
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