Formal Conjugacy Growth in Acylindrically Hyperbolic Groups

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Abstract

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt, and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length conjugacy representatives can be unambiguous context-free.
Original languageEnglish
Pages (from-to)121-157
Number of pages37
JournalInternational Mathematics Research Notices
Volume2017
Issue number1
Early online date14 Apr 2016
DOIs
Publication statusPublished - Jan 2017

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