Growth in free groups (and other stories)—twelve years later

Abstract

We start by studying the distribution of (cyclically reduced) elements of the free groups $F_n$ with respect to their Abelianization (or equivalently, their class in $H_1(F_n, \mathbf{Z})$). 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 $\mod p$ ($p$—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.