The Frattini subgroup for subgroups of hyperbolic groups

TL;DR

This paper proves that the Frattini subgroup of any finitely generated subgroup of a hyperbolic group is finite, revealing a new structural property of these subgroups.

Contribution

It establishes the finiteness of the Frattini subgroup for finitely generated subgroups within hyperbolic groups, a novel result in geometric group theory.

Findings

Finiteness of the Frattini subgroup in finitely generated subgroups of hyperbolic groups

Provides new insights into subgroup structure of hyperbolic groups

Advances understanding of algebraic properties in geometric group theory

Abstract

We prove that for a finitely generated subgroup $H$ of a word-hyperbolic group $G$ the Frattini subgroup $F(H)$ of $H$ is finite.