Hyperbolically embedded subgroups and quasi-isometries of pairs

TL;DR

This paper establishes conditions under which quasi-isometries preserve hyperbolically embedded subgroups and explores implications for acylindrical hyperbolicity's invariance in finitely generated groups.

Contribution

It provides new criteria for the preservation of hyperbolically embedded subgroups under quasi-isometries and examines their impact on acylindrical hyperbolicity invariance.

Findings

Quasi-isometries can preserve hyperbolically embedded subgroups under certain conditions.

Hyperbolic properties are invariant under quasi-isometry for specific classes of groups.

Results apply to the study of acylindrical hyperbolicity and group classification.

Abstract

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