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After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have nice geometric interpretation, I can not help asking the following question:

Question 1: Are there any results in group theory that are much easier to be obtained by geometric, or topolocial tools that by purely algebraic tools?

Of course, the first typical example comes to mind is that any subgroup of a free group is free. Are there any other examples?

Since it is a common idea in group alegbras that we compare properties of groups with the algebra it generated, and we know that free group $F_2=Z*Z$ has a nice geometric interpretation by considering its cayley graph, I would like to ask the following question,

Question 2: Denote $L(F_2)$ to be the von-Neumann algebra associated to the free group generated by two elements, is there any geometric interpertation for this algebra?

Of course, another motivation to ask the 2nd question is that we have already many known candidate invariants associated to these free group factors, such as the free dimension in free probability, the number of generators etc, but in my opinion, most of them are more of less algebraic or analytic in natural.

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Subgroups of free groups can be viewed through things called "Stallings Foldings". This is very convenient. The original paper is called "Topology of finite graphs", and includes very quick proofs that if $A$ and $B$ are finitely generated subgroups of a free group then $A\cap B$ is finitely generated, and that finitely generated subgroups of free groups are closed in the profinite topology. – user1729 Nov 14 '12 at 7:18
Another example I quite like is one can prove a subgroup is malnormal by simply showing that the non-diagonal component of the fibre product is a forest. A Subgroup is malnormal if $H^g\cap H\neq 1\Rightarrow g\in H$, and the non-diagonal components of the fibre product correspond to a change of base point. Again, very neat and very nice. – user1729 Nov 14 '12 at 7:20
(I should have said - foldings are operations on graphs. They use the fact that free groups are the fundamental groups of graphs. No idea why I didn't say that above!) – user1729 Nov 14 '12 at 9:54
@user1729, thanks for the reference. Just curiosity, your usename reminds me of the story of Ramanujan and the taxi number. – ougao Nov 15 '12 at 0:08
Indeed.${}{}{}$ – user1729 Nov 15 '12 at 7:27
up vote 1 down vote accepted

Here are two other examples how topology can be used to obtain results in group theory.

Let $G$ be a torsionfree group. Then it is a classical conjecture that the Whitehead group $Wh(G)$ vanishes. This Whitehead group is a quotient of the first algebraic K-theory group of $\mathbb{Z}G$ and thus a priori purely algebraically defined.

Now by the famous s-cobordism states that if $G = \pi_1(X)$ is the fundamental group of some manifold $X$ then there is a canonical bijection

$$ Wh(G) \cong \{ \text{h-cobordisms from X to some other manifold Y} \}/diffeomorphism $$

and indeed some proofs that $Wh(G)=0$ are obtained by studying the right hand side, and geometrically showing that there are only trivial h-cobordisms.

As a second example, again let $G$ be a torsionfree group. Then it is conjectured that the complex group ring $\mathbb{C}G$ has no nontrivial idempotent elements, i.e., only $0$ and $1$ are such idempotents.

This conjecture for example follows from either the Farrell-Jones Conjecture or the Baum-Connes Conjecture. These conjectures are about algebraic $K$ and $L$ theory and topological $K$-theory for certain group algebras and have been proved for a large class of groups. A nice survey about these conjectures can be found on Prof. Wolfgang Luecks homepage, just google his name. Moreover I do think that these conjectures are of topological nature, (they are related to surgery theory and index theory).

As far as I know proving either one of these conjectures for the group $G$ is one of the main ways to get to the idempotent conjecture, which a priori does not have anything todo with topology.

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