Riemann Surfaces, Random Regular Graphs, and Virtual Permutations
Alexander Gamburd
2.00 pm-4.00 pm May 20, (MONDAY!)
383N (3rd floor lounge) at STANFORD
I will describe recent joint work with Robert Brooks and Eran Makover
concerning the global geometric quantities (in particular, the first
eigenvalue of the Laplacian, injectivity radius, and diameter) of a
"typical" compact Riemann surface of large genus. The approach is based
on compactifying finite-area Riemann surfaces associated with random
cubic graphs; by a theorem of Belyi these are "dense" in the space of
compact Riemann surfaces. The questions in asymptotic enumeration
arising in this approach are related to the model for random regular
graphs introduced by Bollobas, and to the space of virtual permutations
introduced by Kerov, Olshansky, and Vershik.