Growth Rate Gap for Stable Subgroups

TL;DR

This paper proves that in broad classes of groups, stable subgroups have strictly smaller growth rates than the entire group, extending previous results by removing extra assumptions.

Contribution

It establishes a growth gap for stable subgroups in Morse local-to-global groups without requiring torsion-free or residually finite conditions.

Findings

Stable subgroups have strictly less growth rate than the ambient group.

The result applies to a broad class of groups including mapping class groups and CAT(0) groups.

Generalizes previous work by removing additional assumptions.

Abstract

We prove that stable subgroups of Morse local-to-global groups exhibit a growth gap. That is, the growth rate of an infinite-index stable subgroup is strictly less than the growth rate of the ambient Morse local-to-global group. This generalizes a result of Cordes, Russell, Spriano, and Zalloum in the sense that we removed the additional torsion-free or residually finite assumptions. The Morse local-to-global groups are a very broad class of groups, including mapping class groups, CAT(0) groups, closed $3$-manifold groups, certain relatively hyperbolic groups, virtually solvable groups, etc.