Formal Conjugacy Growth in Acylindrically Hyperbolic Groups

Abstract

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.2010 Mathematics Subject Classification: 20F67, 68Q45.

Conjugacy growth in the higher Heisenberg groups

“…A closely related measure of conjugacy growth is the formal power series whose coefficients are the values of the conjugacy growth function. Ciobanu, Hermiller, Holt and Rees [7] and Antolín and Ciobanu [1] confirmed a conjecture of Rivin [29], that the conjugacy growth series of a hyperbolic group is a rational function if and only if the group is virtually cyclic. Continuing in a similar vein, the author proved the rationality of the series for all virtually abelian groups [12], and Gekhtman and Yang proved transcendence for relatively hyperbolic groups and certain acylindrically hyperbolic groups [17].…”

“…Suppose that N = Γ × H D is torsion-free. Then the natural homomorphism N → N/N (1) induces (via Lemma 3.10) an epimorphism θ : Aut(Γ × H D ) → M.…”

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

“…for any n ≥ 1. This property admits several interesting applications, for instance, to statistical hyperbolicity [32] [60], to counting conjugacy classes and automatic structures [1], and to the finiteness of Bowen-Margulis-Sullivan measure [70] [79].…”

The Conjugacy Ratio of Groups

“…For example, the standard and conjugacy growth rates (i.e. taking the limit of the nth root of the function at n) are equal in some of the most frequently encountered families of infinite groups: hyperbolic groups [AC17], graph products [CHM17], = 1, Definition 3. 1.…”

“…For example, the standard and conjugacy growth rates (i.e. taking the limit of the nth root of the function at n) are equal in some of the most frequently encountered families of infinite groups, including hyperbolic groups [1], graph products [5] and many wreath products [11]; thus in these examples the quotient of the two functions, as a function of n, must be at most subexponential, and if the conjugacy ratio is 0, the convergence to 0 will not be very fast.…”