EXPANSION IN LINEAR GROUPS, SPECTRAL GAPS AND APPLICATIONS

This talk is a partial survey of some developments over recent years around the expansion theory ofthe Cayley graphs of linear groups and spectral gaps of the corresponding Hecke operators. A sample resultis the extension of Selberg's theorem for congruence subgroups of SL2(Z) to the setting of arbitrary non-elementary subgroups. Underlying these advances are techniques from arithmetic combinatorics that will be briefly indicated. The main emphasis of this (non-technical) presentation will be rather the applications however. They are quite diverse and relate to a variety of different issues in various fields: hyperbolic lattice point counting, number theory, geometry and questions from computer science.