Growth in free groups (and other stories)

TL;DR

This paper investigates the distribution and growth properties of elements in free groups, deriving explicit formulas and asymptotic behaviors, and extends these techniques to graph cycles and conjugacy class growth.

Contribution

It introduces new methods using Chebyshev polynomials to analyze distributions in free groups and applies these to graph cycles and conjugacy class growth functions.

Findings

Explicit generating function for cyclically reduced elements

Asymptotic equidistribution of reductions mod p

Growth functions for conjugacy classes

Abstract

We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions $\mod p$ ($p$ -- an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.