Transitivity degrees of countable groups and acylindrical hyperbolicity

Abstract

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.

Invariant random subgroups of groups acting on hyperbolic spaces

“…action of G with "small" stabilizers (e.g., amenable, or, more generally, without non-cyclic free subgroups) is essentially free. This statement can be made precise using the notion of a small subgroup introduced in [15].…”

Groups Acting Acylindrically on Hyperbolic Spaces

“…The property of being MIF is much stronger than being identity free and imposes strong restrictions on the algebraic structure of G. For example, if G has a non-trivial center, then it satisfies the non-trivial mixed identity [a, x] = 1, where a ∈ Z(G)\{1}. Similarly, it is easy to show (see [39]) that a MIF group has no finite normal subgroups, is directly indecomposable, has infinite girth, etc. By constructing highly transitive permutation representations of acylindrically hyperbolic groups, Hull and the author proved that every acylindrically hyperbolic group with trivial finite radical is MIF [39].…”

On acylindrical hyperbolicity of groups with positive first ℓ2-Betti number

“…We note that, unlike ordinary hyperbolicity, acylindrical hyperbolicity of finitely generated groups is not known to be a quasi-isometry invariant. For the definition of the first 2 -Betti number of a group, we refer the reader to [8]. Readers who are not familiar with this theory may think of the following Lück's Approximation Theorem as a definition in the particular case of finitely presented residually finite groups.…”

“…By Theorem 2.2, every finitely presented residually finite group G with β is essentially due to Hamenstädt who proved it in different terms (see [6] and the discussion before [13,Theorem 8.3]). For (d), we refer the reader to [8].…”

“…Not satisfying any non-trivial mixed identity is a rather restrictive condition. It implies several interesting model-theoretic, geometric, and algebraic properties of G. For details, see [8].…”