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

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

2011-06-24

arxiv.org/abs/1106.5947v1

Illinois Journal of Mathematics vol 54 (Spring 2010), pp 327-370

Mathematics

Group Theory

45pp, appeared in the Schupp volume of the Illinois Journal of Mathematics, published version of arXiv:math/9911076

Scientific paper

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.

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