Conjugacy growth of commutators

2018/02/26 by Park, Peter S.
#05A16 #11D45 #20E05 #20E06 #20E45 #20F65 #20H10 #37E35 #37F20 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1802.09507

Abstract

For the free group F_r 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 F_r (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 F_r with a given word length.

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