Examples of group-theoretic results more easily obtained through topology or geometry

Earlier, I was looking at a question here about the abelianization of a certain group $X$. Since $X$ was the fundamental group of a closed surface $\Sigma$, it was easy to compute $X^{ab}$ as $\pi_1(\Sigma)^{ab} = H_1(\Sigma)$, then use the usual machinery to compute $H_1(\Sigma)$. That made me curious about other compelling examples of solving purely (for some definition of 'purely') algebraic questions that are accessible via topology or geometry. The best example I can think of the Nielsen-Schreier theorem, which is certainly provable directly but has a very short proof by recasting the problem in terms of the fundamental group of a wedge product of circles. Continuing this line of reasoning leads to things like graphs of groups, HNN-extensions, and other bits of geometric group theory.

What are some other examples, at any level, of ostensibly purely group-theoretic results that have compelling, shorter topological proofs? The areas are certainly closely connected; I'm looking more for what seem like completely algebraic problems that turn out to have completely topological resolutions.

The Kaplansky Conjecture asserts that the group ring $\Bbb Q G$ contains no non-trivial zero-divisors when $G$ is a torsionfree group.

It is implied by the Atiyah Conjecture, (a version of) which states that for every compact connected CW complex $X$ with $\pi_1(X)=G$, all $\ell^2$-Betti numbers are integers.

The Atiyah Conjecture is now known to be true for a great deal of groups - in particular for groups for which the truth of the Kaplansky Conjecture was unknown before.

There is an exact sequence, due to Hopf: $$\pi_2(X) \to H_2(X) \to H_2(\pi_1(X)) \to 0,$$ where the first map is the Hurewicz map. In some sense, $H_2(\pi_1(X))$ measures how surjective the (2-dimensional) Hurewicz map is. The purpose of this answer is to sell you on this exact sequence. Here's a small result you can prove with it.

Theorem: if $G$ is the fundamental group of a homology sphere, then $H_1(G) = H_2(G) = 0$. In particular, this is true of the binary icosahedral group (a double cover of $A_5$), which is the fundamental group of the Poincare sphere.

Proof: $H_1(G) = G^{ab} = H_1(X) = 0$. That $H_2(G) = 0$ follows immediately from the above exact sequence, because $H_2(\pi_1(X))$ is a quotient of $H_2(X)$.

Two notes.

1) I don't know an easier or different way at all to prove this fact.

2) This property actually characterizes fundamental groups of homology spheres. It is an old theorem of Kervaire that if $G$ is a finitely presented group with trivial $H_1$ and $H_2$, then for fixed $n > 4$, you can find a homology $n$-sphere with fundamental group $G$. $n=3,4$ are (as usual!) something of a mystery.