Walks on Free Groups and other Stories -- twelve years later

TL;DR

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

Contribution

It introduces new analytical techniques for studying distributions in free groups and applies them to graph cycles and conjugacy class growth functions.

Findings

Distribution of free group elements is explicitly characterized.

Modulo prime classes are asymptotically equidistributed.

Growth functions of conjugacy classes are analyzed.

Abstract

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. 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 modulo 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.