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

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, Subgroup Growth of Fuchsian Groups, and Random Walks

Random Walks on Co-compact Fuchsian Groups

It is proved that the Green’s function of a symmetric finite range randomwalk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence R. It is also shown that Ancona’s inequalities extend to R, and therefore that the Martin boundary for R−potentials coincides with the natural geometric boundary S, and that the Martin kernel is uniformly Hölder continuous. Fi...

Alternating Quotients of Fuchsian Groups

It all started with a theorem of Miller [14]: the classical modular group PSL2Z‘ has among its homomorphic images every alternating group, except A6; A7; and A8. In the late 1960s Graham Higman conjectured that any (finitely generated non-elementary) Fuchsian group has among its homomorphic images all but finitely many of the alternating groups. This reduces to an investigation of the cocompac...

Random Walks on Finite Groups

Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as t...