  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.
