Diophantine properties of groups of toral automorphisms

This dissertation studies 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 non-commutative Diophantine approximation problem. The methods require construction of spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, similar estimates for this problem in higher dimension can be obtained by using harmonic analysis.