# Transitivity degrees of countable groups and acylindrical hyperbolicity

@article{Hull2015TransitivityDO, title={Transitivity degrees of countable groups and acylindrical hyperbolicity}, author={Michael Hull and Denis V. Osin}, journal={Israel Journal of Mathematics}, year={2015}, volume={216}, pages={307-353} }

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…

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