Transitivity degrees of countable groups and acylindrical hyperbolicity

By Michael Hull, Denis V. Osin. 2015

49 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b7ba41c3a0eee18430c4f523588375f4e870be5b

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In Computational Theory and Mathematics

Denis V. Osin

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, background): “Similarly, it is easy to show (see [39]) that a MIF group has no ﬁnite normal subgroups, is directly indecomposable, has inﬁnite girth, etc.”