Bounded subgroups of relatively finitely presented groups
type: Item
instance of: scholarly article
title: Bounded subgroups of relatively finitely presented groups
author: Eduard Schesler
published in: Algebraic \& Geometric Topology
publication date: 2024-08-26T00:00:00Z
review text: The notion of a group \(G\) that is hyperbolic relative to a finite set \(H_{\Lambda}\) of its subgroups was introduced by \textit{M. Gromov}, in [Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)], as a generalization of a word hyperbolic group. In that definition, the groups \(H\in H_{\Lambda}\) are stabilizers of points at infinity of a certain hyperbolic space \(X\) the group \(G\) acts on.\N\NLet \(G\) be a finitely generated group that is relatively finitely presented with respect to a collection \(H_{\Lambda}\) of peripheral subgroups such that the corresponding relative Dehn function is well-defined. In the paper under review, the author proves that every infinite subgroup \(H\) of \(G\) that is bounded in the relative Cayley graph of \(G\) with respect to \(H_{\Lambda}\) is conjugate into a peripheral subgroup. As an application, he obtains a trichotomy for subgroups of relatively hyperbolic groups. Furthermore, he proves the existence of the relative exponential growth rate for all subgroups of limit groups.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 7901546
zbMATH Keywords: hyperbolic group
zbMATH Keywords: relatively finitely presented group
zbMATH Keywords: peripheral subgroup
zbMATH Keywords: Cayley graph
zbMATH Keywords: growth rate
zbMATH Keywords: Dehn function
MaRDI profile type: Publication
cites work: A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS
cites work: Relatively hyperbolic groups.
cites work: Cogrowth and amenability of discrete groups
cites work: Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. (Geometry and group theory. The hyperbolic groups of Gromov)
cites work: Growth gap in hyperbolic groups and amenability
cites work: Combination of convergence groups.
cites work: Growth of quasiconvex subgroups
cites work: Relatively hyperbolic groups
cites work: Q3930365
cites work: Q3772406
cites work: Dehn filling in relatively hyperbolic groups.
cites work: Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan)
cites work: A note on curvature and fundamental group
cites work: Subnormal subgroups in free groups, their growth and cogrowth
cites work: Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems
cites work: Acylindrically hyperbolic groups
cites work: The relative exponential growth rate of subgroups of acylindrically hyperbolic groups
cites work: Diophantine geometry over groups. I: Makanin-Razborov diagrams
cites work: Relative growth series in some hyperbolic groups
DOI: 10.2140/AGT.2024.24.1551
description: scientific article; zbMATH DE number 7901546