Intersection properties of stable subgroups and bounded cohomology

TL;DR

This paper investigates the intersection properties of stable subgroups in finitely generated groups, establishing finiteness results and characterizing extendable quasimorphisms via bounded cohomology.

Contribution

It introduces new intersection properties of stable subgroups and characterizes conditions for extending quasimorphisms in hyperbolically embedded subgroups.

Findings

Finite height, width, and bounded packing of stable subgroups.

Characterization of extendable quasimorphisms based on intersection properties.

Insights into bounded cohomology and hyperbolically embedded subgroups.

Abstract

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.