TITLE: Examples of group-theoretic results more easily obtained through topology or geometry
QUESTION [53 upvotes]: 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.

REPLY [5 votes]: One influence of geometry and topology on group theory has been to extend group theoretic methods. Philip Higgins described to me how he first thought of using groupoids by reading about covering spaces in the book on Homology Theory  by Hilton and Wylie,  and realising that the account of covering spaces was all about groupoids. You can read about the applications he found in the 1971 book Categories and Groupoids (downloadable); this gives groupoid proofs of subgroup theorems mentioned on this page. Note that groupoids allow notions of fibration and covering morphisms, modelling topological notions. 
An application of higher groupoids to homotopy theory which Loday and I published in 1984 led to  a nonabelian tensor product of groups which act on each other in a "compatible" way. A simple example is two normal subgroups $M,N$ of a group $P$. The commutator map $[\,, \,]:M \times N \to P, (m,n) \mapsto mnm^{-1}n^{-1}$ has properties of $[mm',n], [m,nn']$,  which make it, not bimultiplicative, but a biderivation. The universal biderivation is written $M \times N \to M \otimes N$, so the commutator map factors through a morphism of groups $\kappa: M \otimes N \to P$. Graham Ellis proved that $M \otimes N $ is finite if $M,N$ are finite.
There are applications of this tensor product to the homology of groups. For example if $1 \to R \to F \to P\to 1$ is an exact sequence with $F$ free, then $$H_3(P) \cong \text{Ker} (R  \wedge F \to P)$$ where $R \wedge F$ is the quotient of $R \otimes F$ by the relations $r \otimes r =1$ for $r \in R$; and to algebraic topology: for example $$\pi_3S(K(P,1))\cong \text{Ker} (\kappa: P \otimes P \to P),$$ where $S$ is suspension. A number of group theorists have taken up the ideas, particularly on calculating $P \otimes P$ for classes of groups, see this bibliography, which has over 170 items, including a description  dating from 1952 of $H_2(P)$ essentially as the kernel of $P \wedge P \to P$.