Representation Growth and Rational Singularities

Abstract: Representation Growth and Rational Singularities Application # 2012247 Avraham Aizenbud (Weizmann Institute) Nir Avni (Northwestern University) November 1, 2012 We propose a research on the interface of two branches of algebra: asymp-totic group theory on the one hand and combinatorial algebraic geometry on the other. More speci�cally, within group theory we concentrate on the asymptotics of the number of representations of arithmetic lattices and pro-finite groups. The study of such asymptotics is called representation growth. Within algebraic geometry, we concentrate on the singularities of certain varieties. The varieties of interest are deformation varieties of surface groups inside linear algebraic groups, i.e., varieties of the form Hom(�1�;G), where � is a compact two dimensional surface, and G is an algebraic group. In previous work, we found a connection between representation growth of p- adic analytic groups and some homological invariants of the singularities of associated deformation varieties. In this project, we will try to generalize the connection between representation growth and singularities of deformation varieties. One direction is to generalize the connection to positive characteristic; another direction is to generalize the connection to arithmetic lattices. A second goal of this project is to study the singularities of deformation varieties. We propose to study these deformation varieties by degenerating them into simpler varieties. These degenerations fall into a new class of algebraic varieties that are build from (combinatorial) graphs and symplectic vector spaces. We intend to study the structure of varieties in this class in detail.