Random Walks on Co-compact Fuchsian Groups

Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth an...

Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks

We survey a number of powerful recent results concerning diophantine and asymptotic properties of (ordinary) characters of symmetric groups. Apart from their intrinsic interest, these results are motivated by a connection with subgroup growth theory and the theory of random walks. As applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, as well as a finite...

Character Theory of Symmetric Groups, Subgroup Growth of Fuchsian Groups, and Random Walks

Ž . Splitting Groups of Signature

Let G be a co-compact Fuchsian group. As such, it acts discontinuously on the hyperbolic plane H and the quotient is a closed hyperbolic 2-orbifold. The structure theory for co-compact Fuchsian groups yields a canonical presentation for such groups. This information is described by the signature s s g ; m , . . . , m 1 Ž . Ž . 1 r where g is the genus of the base surface of the quotient 2-orbif...

Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the boundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give an affirmative answer to this question, by considering the behavior of the word metric along typical geodesic rays and their excursions into cusps. We also consider the b...