Commensurated hyperbolic subgroups

TL;DR

This paper investigates the structure of certain hyperbolic subgroups within hyperbolic groups, revealing their virtual decompositions and splitting properties, which advances understanding of hyperbolic group subgroup dynamics.

Contribution

It establishes that non-elementary hyperbolic commensurated subgroups of infinite index are virtually free products of surface groups and free groups, and describes splitting over 2-ended subgroups in hyperbolic bundles.

Findings

Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.

One-ended hyperbolic groups as fibers in hyperbolic bundles virtually split over 2-ended subgroups.

Provides structural insights into hyperbolic subgroup behavior and splitting properties.

Abstract

We show that if H is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group G, then H is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group H is a fiber of a non-trivial hyperbolic bundle then H virtually splits over a 2-ended subgroup.