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Spring 2010 Growth in free groups (and other stories)—twelve years later
Igor Rivin
Illinois J. Math. 54(1): 327-370 (Spring 2010). DOI: 10.1215/ijm/1299679752


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.


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Igor Rivin. "Growth in free groups (and other stories)—twelve years later." Illinois J. Math. 54 (1) 327 - 370, Spring 2010.


Published: Spring 2010
First available in Project Euclid: 9 March 2011

zbMATH: 1225.05128
MathSciNet: MR2776999
Digital Object Identifier: 10.1215/ijm/1299679752

Primary: 05C20‎ , 05C25 , 05C38 , 42A05 , 60F05 , 60J10
Secondary: 22E27

Rights: Copyright © 2010 University of Illinois at Urbana-Champaign


Vol.54 • No. 1 • Spring 2010
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