# 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.