Formal conjugacy growth in acylindrically hyperbolic groups

TL;DR

This paper proves that non-elementary hyperbolic groups have transcendental conjugacy growth series, confirming Rivin's conjecture, and extends the analysis to acylindrically hyperbolic groups, showing limitations on their conjugacy representatives.

Contribution

It confirms Rivin's conjecture for hyperbolic groups and explores formal language constraints in acylindrically hyperbolic groups, providing new insights into their conjugacy growth.

Findings

Conjugacy growth series of non-elementary hyperbolic groups is transcendental.

No unambiguous context-free set of minimal conjugacy representatives exists for acylindrically hyperbolic groups.

Variations of Rivin's conjecture are established for commensurability and primitive classes.

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.