Vladimir Finkelshtein

Diophantine approximation problems for groups of toral automorphisms.

I will present sharp rates for a shrinking target problem for the action of arbitrary subgroups 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.