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Journal of the London Mathematical Society Advance Access originally published online on January 12, 2007
Journal of the London Mathematical Society 2007 75(1):47-66; doi:10.1112/jlms/jdl006
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© 2007 London Mathematical Society

Random walks on free products of cyclic groups

Jean Mairesse

LIAFA
CNRS – Université Paris 7
Case 7014,
2, Place Jussieu,
75251 Paris cedex 05
France
jean.mairesse{at}liafa.jussieu.fr

Frédéric Mathéus

LMAM
Université de Bretagne-Sud
Campus de Tohannic, BP 573
56017 Vannes
France
frederic.matheus{at}univ-ubs.fr

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitly solved to get closed form formulae for the drift. The examples considered are Z/2Z{star} Z/3Z, Z/3Z{star} Z/3Z, Z/kZ{star} Z/kZ and the Hecke groups Z/2Z{star} Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy and the growth of the group.

2000 Mathematics Subject Classification 60J10, 60B15, 60J22, 65C40 (primary), 28D20, 37M25 (secondary).

Received March 31, 2004; revised May 10, 2006; published online January 12, 2007.


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