Regularity of Morse geodesics and growth of stable subgroups

TL;DR

This paper demonstrates that Morse local-to-global groups exhibit exponential growth rates surpassing their infinite index stable subgroups, extending known results to a broader class of groups including mapping class groups and CAT(0) groups.

Contribution

It introduces a theory of automatic structures on Morse geodesics, generalizing growth and subgroup properties in Morse local-to-global groups beyond hyperbolic groups.

Findings

Morse groups grow exponentially faster than stable subgroups

Stable subgroups can be characterized by regular languages

Morse boundary points are dense and contained in limit sets

Abstract

We prove that Morse local-to-global groups grow exponentially faster than their infinite index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds. To accomplish this, we develop a theory of automatic structures on Morse geodesics in Morse local-to-global groups. Other applications of these automatic structures include a description of stable subgroups in terms of regular languages, rationality of the growth of stable subgroups, density in the Morse boundary of the attracting fixed points of Morse elements, and containment of the Morse boundary inside the limit set of any infinite normal subgroup.