  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.
