Regularity of Morse geodesics and growth of stable subgroups
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.
Keywords
Cite
@article{arxiv.2008.06379,
title = {Regularity of Morse geodesics and growth of stable subgroups},
author = {Matthew Cordes and Jacob Russell and Davide Spriano and Abdul Zalloum},
journal= {arXiv preprint arXiv:2008.06379},
year = {2022}
}
Comments
Version to appear in Journal of Topology. Minor updates, including a new corollary on the non-existance of a Cannon--Thurston map from the boundary of a hyperbolic normal subgroup to the Morse boundary of the ambient Morse local-to-global group