Bounded subgroups of relatively finitely presented groups

Schesler, Eduard

doi:10.2140/agt.2024.24.1551

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Bounded subgroups of relatively finitely presented groups

Eduard Schesler

Algebraic & Geometric Topology 24 (2024) 1551–1567

DOI: 10.2140/agt.2024.24.1551

Abstract

Let $G$ be a finitely generated group that is relatively finitely presented with respect to a collection $H_{Λ}$ of peripheral subgroups such that the corresponding relative Dehn function is well defined. We prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ with respect to $H_{Λ}$ 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.

Keywords

relatively finitely presented groups, relatively hyperbolic groups, growth of groups, coarse geometry

Mathematical Subject Classification

Primary: 20F67

References

Publication

Received: 18 March 2022

Revised: 1 February 2023

Accepted: 1 March 2023

Published: 28 June 2024

Authors

Eduard Schesler
Fakultät für Mathematik und Informatik FernUniversität in Hagen Hagen Germany

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Volume 24, issue 3 (2024)