  We prove sharp estimates in a shrinking target problem for the action of an
arbitrary subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ 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.
