Bounded subgroups of relatively finitely presented groups

TL;DR

This paper investigates the structure of subgroups within relatively finitely presented groups, showing bounded subgroups are conjugate into peripheral subgroups, and explores subgroup classifications and growth rates in related group classes.

Contribution

It establishes that bounded subgroups are conjugate into peripheral subgroups and introduces a subgroup trichotomy and exponential growth rate results for limit groups.

Findings

Bounded subgroups are conjugate into peripheral subgroups.

A trichotomy for subgroups of relatively hyperbolic groups.

Existence of relative exponential growth rates for all subgroups of limit groups.

Abstract

Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is conjugate into a peripheral subgroup. As an application, we obtain a trichotomy for subgroups of relatively hyperbolic groups. Moreover we prove the existence of the relative exponential growth rate for all subgroups of limit groups.