Diophantine properties of groups of toral automorphisms

We prove sharp estimates in a shrinking target problem for the action of an arbitrary subgroup $\mathrm{\Gamma }$ of $S{L}_2(\mathbb{Z})$ on the 2-torus. This can also be viewed as a non-commutative Diophantine approximation problem. The methods require constructing spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, using Fourier analysis we give estimates for the same problem in higher dimensions.