Conjugacy growth of commutators

Abstract

For the free group Fr on r > 1 generators (respectively, the free product G1 * G2 of two nontrivial finite groups G1 and G2), we obtain the asymptotic for the number of conjugacy classes of commutators in Fr (respectively, G1 * G2) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of G1 and G2). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy classes of commutator-subgroup elements in Fr with a given word length.

Commutators in $SL_2$ and Markoff surfaces I

“…Wicks' theorem [28] describes explicitly which elements are commutators in such a free product, in terms of their spelling as words in the generators. As shown in [18], if one orders the {Z} Γ 's by their minimal word length, then very few of these are commutators; roughly squareroot of the total number. According to Theorem 2.4, the failure of the typical Z to be a commutator is witnessed in some finite quotient of Γ .…”

“…In [17] this procedure was implemented for |t| < 1000. He finds that for most admissible t's, T t (Z) is not empty, and typically only a few of the h(t) classes are commutators.…”

Combinatorial growth in the modular group

“…The study of asymptotic growth rates of geometric lengths of various classes of closed geodesics has a long and storied history beginning with Huber's result for all closed geodesics, to Mirzakhani's growth rate of the simple closed geodesics, to more general results for non-simple closed geodesics and reciprocal geodesics [1-4, 8, 9, 21, 23, 27]. Concurrently there is the study of such geodesics in terms of word length or equivalently primitive conjugacy classes and their word length growth rates leading to more abstract, algebraic investigations of groups such as surface groups or free groups [5,7,16,22,25,26,29]. Papers involving normal forms, enumeration schemes for curves, Farey arithmetic, and generating elements in a non-abelian free group include [10][11][12][13][14][15]28].…”

Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

“…Wicks' theorem [Wic62] describes explicitly which elements are commutators in such a free product, in terms of their spelling as words in the generators. As shown in [Par19], if one orders the {Z} Γ 's by their minimal word length, then very few of these are commutators; roughly squareroot of the total number. According to Theorem 2.4, the failure of the typical Z to be a commutator is witnessed in some finite quotient of Γ .…”