Growth Rate Gap for Stable Subgroups

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.