Frédéric Mathéus

Growth series for Artin groups of dihedral type

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = 〈a,b | prod(a,b;k) = prod(b,a;k)〉 where prod(s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generators {a,b,a-1, b-1} are rational. We provide explicit formulas for the series.

Randomly growing braid on three strands and the manta ray.

Consider the braid group B3 = and the nearest neighbor random walk defined by a probability \nu with support {a,b,a^-1,b^-1}. The rate of escape of the walk is explicitely expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitely describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

Random Walks on Groups With a Tree-Like Cayley Graph

We consider a transient nearest neighbor random walk on a group G with finite set of generators E.The pair (G, E) isassumed to admit a naturalnotion of normal form words which are modified only locally when multiplied by generators. The basic examples are the free products of a finitely generated free group and a finite family of finite groups,with natural generators. We prove that the harmonic measure is Markovian and can be completely described via afinite set of polynomial equations. It enables to compute the drift,the entropy,the probability of ever hitting an element,and the minimal positive harmonic functions of the walk. The results extend to monoids. In several simple cases of interest,the set of polynomial equations can be explicitly solved, toget closed form formulas for the drift,the entropy,… Various examples are treated:the modular group Z/2Z * Z/3Z,the Hecke groups Z/2Z * Z/kZ,the free products of two isomorphic cyclic groups Z/kZ*Z/k7G,the braid group B3,and Artin groups with two generators.

Empilements de cercles et discrétisation quasiconforme: comportement asymptotique des rayons

Abstract. In a discretization scheme for conformal mappings with circle packings previously considered by Colin de Verdière and the author, a two-term asymptotic expansion of the radii of the image circles is obtained, with a control on the error. The vanishing of the second term characterizes Doyle spirals.

Sharp lower bounds for the asymptotic entropy of symmetric random walks

The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.