Illinois Journal of Mathematics

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

Igor Rivin

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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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Illinois J. Math., Volume 54, Number 1 (2010), 327-370.

First available in Project Euclid: 9 March 2011

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Zentralblatt MATH identifier

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C20: Directed graphs (digraphs), tournaments 05C38: Paths and cycles [See also 90B10] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F05: Central limit and other weak theorems 42A05: Trigonometric polynomials, inequalities, extremal problems
Secondary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)


Rivin, Igor. Growth in free groups (and other stories)—twelve years later. Illinois J. Math. 54 (2010), no. 1, 327--370. doi:10.1215/ijm/1299679752.

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