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 $\infty$. 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.

References in corpus (8)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Highly Transitive Actions of Out(Fn)
• Small cancellation in acylindrically hyperbolic groups
• Acylindrically hyperbolic groups
• Actions of Maximal Growth
• Acylindrical hyperbolicity of groups acting on trees
• On Residualizing Homomorphisms Preserving Quasiconvexity
• Highly transitive actions of groups acting on trees

Cited by in corpus (5)

• Maximal von Neumann subalgebras arising from maximal subgroups
• Automorphisms of algebraic varieties and infinite transitivity
• Generic IRS in free groups, after Bowen
• Highly faithful actions and dense free subgroups in full groups
• Invariant random subgroups of groups acting on hyperbolic spaces