2606.13632Jun 11, 2026In this new version, we added a combinatorial proof for a sphere expansion property in hyperbolic groups, as well as adding references

Growth of Approximate Groups in Hyperbolic Groups

Michael Saks, Gal Yehuda

Group Theory Combinatorics

arXiv PDF

Abstract

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If $G$ is a finitely generated hyperbolic group and $A\subseteq G$ is infinite with $A^2\subseteq AX$ for some finite $X\subseteq G$

Briefing

What this paper is actually doing

Sign in and connect a model key to generate a briefing for this paper.

Socratic margin

Ask the abstract

Questions are grounded in the title and abstract. If the PDF is required, arXivDesk will say so.

Sign in to ask questions about this paper.

Nearby in the stack

math/0504212

over 21 years ago

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

V. Metaftsis, M. Sykiotis

Group Theory Geometric Topology

1107.1826

about 15 years ago

Conjugacy growth of finitely generated groups

M. Hull, D. Osin

Group Theory Geometric Topology

1310.8605

almost 13 years ago

Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Yago Antolin, Ashot Minasyan, Alessandro Sisto

Group Theory Geometric Topology

1404.4335

over 12 years ago

Growth Tight Actions of Product Groups

Christopher H. Cashen, Jing Tao

Group Theory

1103.0114

over 15 years ago

Embeddings of SL(2,Z) into the Cremona group

Jérémy Blanc, Julie Déserti

Algebraic Geometry Group Theory

2608.13414

3 days ago

Synthetic Buildings for Finite Groups

Emily Gullerud, Peter Webb

Group Theory Algebraic Topology Representation Theory

, then either

\langle A\rangle

is virtually cyclic, or

A

has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if