Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:

> There does not exist any Riemannian metric on the torus whose sectional curvature is $<0$.

This is a consequence of Milnor's article *A note on curvature and fundamental group*, where he proves that the fundamental group of a negatively-curved Riemannian manifold has exponential growth. On the other hand, the fundamental group of the torus, namely $\mathbb{Z}^2$, has quadratic growth.

> Any compact Riemannian manifold whose sectional curvature is $\equiv 0$ has a torus as a finite cover.

This is a consequence of Bieberbach theorem. 

More recently, showing that quasiconvex subgroups of hyperbolic cubulable groups are separable was the key point in the proof of the virtual Haken's conjecture. However, this is more technical.