The whole field of geometric group theory very much uses geometric and topological arguments to prove group theory facts. Stallings for example used topological arguments to prove that only free groups have cohomological dimension 1. He also gave a topological proof of Grushko's theorem on generators of free products.
Gromov's proof that only virtually nilpotent groups have polynomial growth uses geometry and topology.
The train track proof of Scott's conjecture that the fixed point set of a free group automorphism has rank n is topological.
Van Kampen diagram proofs line Lyndons theorem on cohomlogical dimension of one relator groups tend to use some topology plus properties of planar graphs.
