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}
}
  • M. Hull, D. Osin
  • Published 17 January 2015
  • Mathematics
  • Israel Journal of Mathematics
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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References

SHOWING 1-10 OF 64 REFERENCES

Properties of acylindrically hyperbolic groups and their small cancellation quotients

We investigate the class of acylindrically hyperbolic groups, which includes many examples of groups which admit natural actions on hyperbolic metric spaces, such as hyperbolic and relatively

Lacunary hyperbolic groups

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R‐tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying

Ordering the space of finitely generated groups

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets

Small cancellations over relatively hyperbolic groups and embedding theorems

We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable

Small cancellation in acylindrically hyperbolic groups

We generalize a version of small cancellation theory to the class of acylindrically hyperbolic groups. This class contains many groups which admit some natural action on a hyperbolic space, including

Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a

Highly transitive representations of free groups and free products

A permutation group is highly transitive if it is n–transitive for every positive integer n. A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every

Highly transitive actions of groups acting on trees

We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is

Acylindrical hyperbolicity of groups acting on trees

We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, one-relator groups,

On Residualizing Homomorphisms Preserving Quasiconvexity #

ABSTRACT H is called a G-subgroup of a hyperbolic group G if for any finite subset M ⊂ G there exists a homomorphism from G onto a non-elementary hyperbolic group G 1 that is surjective on H and
...