Correction to: Acylindrical hyperbolicity of groups acting on trees

Abstract

In Lemma 3.9 of [2], we stated that acylindrical hyperbolicity of a group is invariant under commensurability up to finite kernels (for definitions and background material we refer to [2,4]). This lemma was used to prove some of the main results in [2] and the subsequent paper [4]. Unfortunately, its proof contains a gap. The goal of this erratum is to point out the gap and correct the statements and proofs of results of [2,4] affected by it. The arXiv versions of the papers [2,4] will be updated accordingly. 1.1 The gap The arguments given in [2] correctly prove some parts of Lemma 3.9, which are summarized below. Lemma 1 Let H be an acylindrically hyperbolic group. Suppose that G is a finite index subgroup of H , or a quotient of H modulo a finite normal subgroup, or an extension of H with finite kernel. Then G is also acylindrically hyperbolic. However, we do not know the answer to the following.

Acylindrical hyperbolicity of automorphism groups of infinitely ended groups

“…The price to pay in the conclusion is that without the Nielsen realization assumption, we can only prove that Aut(G) contains a finite-index subgroup which is acylindrically hyperbolic. The stability of acylindrical hyperbolicity under finite-index overgroups is still open to our knowledge (see the discussion in [55]), so a priori we cannot conclude that the entire automorphism group is acylindrically hyperbolic. Nevertheless, many of the interesting properties which can be deduced from being acylindrically hyperbolic, such as the existence of uncountably many normal subgroups [21,Theorem 2.33], pass from a finite-index subgroup to the overgroup, so proving that a group is virtually acylindrically hyperbolic remains of interest.…”

“…Finally, it was proved in [16,Corollary 4.3] that F n ϕ Z is virtually acylindrically hyperbolic if and only if the image of ϕ in Out(F n ) has infinite order. But to our knowledge, it is currently unknown whether virtually acylindrically hyperbolic groups are acylindrically hyperbolic (see [55]), so Corollary 1.5 is new to our knowledge.…”

The Cremona Group and Its Subgroups

“…(2) Minosyan and Osin note that if the answer to this question is yes, the results of [DGO17] allow to give a new proof of the non-simplicity of Bir(P 2 C ) (see [MO15,MO19]).…”

Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

“…The price to pay in the conclusion is that without the Nielsen realisation assumption, we can only prove that Aut(G) contains a finite-index subgroup which is acylindrically hyperbolic. The stability of acylindrical hyperbolicity under finite-index overgroups is still open to our knowledge (see the discussion in [MO19]), so a priori we cannot conclude that the entire automorphism group is acylindrically hyperbolic. Nevertheless, many of the interesting properties which can be deduced from being acylindrically hyperbolic, such as the existence of uncountably many normal subgroups [DGO17, Theorem 2.33], pass from a finite-index subgroup to the overgroup, so proving that a group is virtually acylindrically hyperbolic remains of interest.…”

“…Finally, it was proved in [BK16, Corollary 4.3] that F n ϕ Z is virtually acylindrically hyperbolic if and only if ϕ has infinite order. But to our knowledge, it is currently unknown whether virtually acylindrically hyperbolic groups are acylindrically hyperbolic (see [MO19]), so Corollary 1.5 is new to our knowledge.…”