Regularity of Morse geodesics and growth of stable subgroups

Matthew Cordes, Jacob Russell, Davide Spriano, Abdul Zalloum

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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.
Original languageEnglish
Pages (from-to)1217-1247
Number of pages31
JournalJournal of Topology
Issue number3
Early online date24 Jun 2022
Publication statusPublished - Sept 2022


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