Growth tightness of quotients by confined subgroups

TL;DR

This paper proves that quotients by confined subgroups in certain groups with specific geometric actions exhibit growth tightness, meaning they cannot have purely exponential growth, with implications for recurrent subgroups.

Contribution

It establishes the growth tightness of quotients by confined subgroups in groups with statistically convex-cocompact actions and contracting elements, extending understanding of their geometric properties.

Findings

Quotients by confined subgroups are growth tight in these groups.

The result is sharp; actions cannot be relaxed to purely exponential growth.

Applications to uniformly recurrent subgroups are provided.

Abstract

In this paper, we establish the growth tightness of the quotient by confined subgroups in groups admitting the statistically convex-cocompact action with contracting elements. The result is sharp in the sense that the actions could not be relaxed with purely exponential growth. Applications to uniformly recurrent subgroups are discussed.