Title: Diophantine approximation problems for groups of toral automorphisms

Speaker: Vladimir Finkelshtein (University of Illinois at Chicago)

Abstract: I will present sharp rates for a shrinking target problem for the action of an arbitrary subgroup of SL(2,Z) on the 2-torus. This can also be viewed as a noncommutative Diophantine approximation problem. The methods require construction of spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, I will explain how similar estimates for this problem in higher dimension can be obtained using harmonic analysis.