Let G be a group and H subgroup of G. The notion being hyperbolically embedded in was introduced by Dahmani, Guiraldel, Osin. This note introduces an equivalent definition hyperbolic based on Bowditch’s approach to relatively groups terms fine graphs.

For any field K, we determine the quasi-simple subgroups G of the Chevalley group F4ðKÞ which are generated by a class S of so-called abstract transvection subgroups of G such that any member of S is contained in a long root subgroup of F4ðKÞ. First, we construct a polar space with point set S, which is embedded in the F4-geometry. This yields that a conjugate of G is contained in a classical s...

We investigate some product structures in R. Thompson’s group V , primarily by studying the topological dynamics associated with V ’s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two gen...

Abstract We give technical conditions for a quasi-isometry of pairs to preserve subgroup being hyperbolically embedded. consider applications the and commensurability invariance acylindrical hyperbolicity finitely generated groups.