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We prove that every countable acylindrically hyperbolic group admits a highlytransitive action with finite kernel. This theorem uniformly generalizes manypreviously known results and allows us to answer a question of Garion andGlassner on the existence of highly transitive faithful actions of mappingclass 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 supremumof transitivity degrees of its faithful permutation representations. Further,for any countable group $G$ admitting a highly transitive faithful action, weprove the following dichotomy: Either $G$ contains a normal subgroup isomorphicto the infinite alternating group or $G$ resembles a free product from themodel theoretic point of view. We apply this theorem to obtain new resultsabout universal theory and mixed identities of acylindrically hyperbolicgroups. Finally, we discuss some open problems.

Transitivity degrees of countable groups and acylindrical hyperbolicity