Growth of quasiconvex subgroups

TL;DR

This paper demonstrates that non-elementary hyperbolic groups exhibit exponential growth rates surpassing those of their infinite index quasiconvex subgroups, extending results to relatively hyperbolic and cubulated groups using advanced mathematical tools.

Contribution

It establishes a new growth comparison theorem for hyperbolic groups and extends it to broader classes using growth tightness and rotating families techniques.

Findings

Hyperbolic groups grow exponentially faster than their quasiconvex subgroups.

The main result is extended to relatively hyperbolic and cubulated groups.

Uses automatic structures and Perron-Frobenius theory in proofs.

Abstract

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel, and Osin on rotating families.