Growth Series and Random Walks on Some Hyperbolic Graphs

Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate "holly trees", a family of reduced loops in these graphs. We then apply Grigorchuk's result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.Comment: 21 pages. to appear in monash. mat

Amenability of groups and GG-sets

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups

Growth Series And Random Walks On Some Hyperbolic Graphs

. We calculate several growth series for the graphs X`;m coming from the regular tesselations of the hyperbolic plane considered by Floyd and Plotnick in [FP87]. This family of graphs includes the Cayley graphs of the fundamental groups of surfaces for their natural generating systems. Using a computational framework we call decomposition rules, we extend some results by Cannon and Wagreich [CW92] to these graphs: their growth series are rational functions whose denominator is a Salem polynomial. We derive some regularity properties of the growth coefficients. As an application of our computations we apply Grigorchuk's theory relating cogrowth and random walks [Gri78] to obtain lower estimates for the spectral radius of the Markov operator associated with a symmetric random walk on the graph X`;m . 1. Introduction We consider the graphs X `;m introduced by Floyd and Plotnick in [FP87]. These graphs are `-regular and are the 1-skeleton of a tesselation of the sphere (if (` \Gamma 2)(m..

Generalized Kaloujnine groups, uniseriality and height of automorphisms

We show that the Lie action of the Kaloujnine group K(p, n) on the vector space (Fp)(pn) is uniserial. Using some Radon transform techniques we derive a formula for the height of the elements in K(p, n). A generalization of the Kaloujnine groups is introduced by considering automorphisms of a spherically homogeneous tree. We observe that uniseriality fails to hold for these groups and determine their lower central series; finally we discuss in detail Kaloujnine's description of the characteristic subgroups in terms of the (normal) "parallelotopic" subgroups