If you want to study a group , try to realize as the fundamental group of a topological space. This works best when is infinite and discrete, especially if is finitely presented and torsion-free.
Actions are a good way of studying a group . The actions of by linear transformations on a vector space are the subject of representation theory, for instance. Topological spaces are another particularly fruitful object for actions. Of course, there are many different sorts of topological space and many different sorts of action that one might want to study, depending on the group in question. One common hypothesis is that the group G should act freely and properly discontinuously on the topological space —for brevity, for the remainder of this article such actions shall be referred to as geometric.
If is the fundamental group of a reasonably nice topological space then acts geometrically on the universal cover, and conversely if admits such an action on a space then the quotient space has fundamental group . So the study of geometric actions is equivalent to the study of fundamental groups. Indeed, the construction of an Eilenberg–Mac Lane Space provides such a space for any group, and better still the construction is functorial.
Using this idea, one aims to study a group by finding a particularly nice topological space on which acts, or equivalently by exhibiting as the fundamental group of a nice space . Conversely, attractive hypotheses on impose restrictions on . For instance, if is a CW-complex with finite one-skeleton then is finitely generated (in particular countable), and if has finite two-skeleton then is finitely presented. (This is an equivalent way of looking at group presentations. Any presentation for describes a two-complex with fundamental group —the generators determine the one-skeleton and the relations the two-skeleton.) It is easy to prove these facts using the Seifert–-van Kampen Theorem.
If is aspherical then the homology and cohomology of are equal to the group homology and cohomology of , so if is also compact then a variety of other conditions are imposed including that is torsion-free. Therefore, if we want to be very nice—compact and aspherical—then will have to be finitely presented and torsion-free. (If the torsion-free hypothesis is too onerous, one approach is to remove the requirement that the action of on be free. In this case the resulting quotient is best not thought of as just a space, but rather as a space with some extra structure.) So we have come to the following precept.
If you are interested in a group , try to find a nice space with fundamental group . This is likely to work particularly well if is finitely presented and torsion-free.
These ideas apply very nicely to free groups.
In this example we will give a very simple proof of the Nielsen–Schreier Theorem, which asserts that every subroup of a free group is free, by exhibiting free groups as the fundamental groups of graphs.
By a graph we mean a connected, 1-dimensional CW-complex. In particular, we allow multiple edges (1-cells) between pairs of vertices (0-cells) and also loops—edges that adjoin only one vertex (although such phenomena can be removed by subdividing).
A graph with just one vertex and edges is called a rose with petals. (Here need not be finite.) The Seifert–van Kampen Theorem implies that the fundamental group of a rose with petals is precisely the free group on generators. More generally, let be an arbitrary graph and let be a maximal tree in . Then is a rose and the quotient map is a homotopy equivalence. This proves the following.
The Nielsen–Schreier Theorem follows immediately from this and elementary covering-space theory.
Let be a non-abelian free group and let be a subgroup. Let be a rose such that is the fundamental group of . By standard covering space theory, there is a covering space of with fundamental group . But a covering space of a graph is a graph, so is free. This completes the proof.
Indeed, these techniques work so well for free groups that a large proportion of the modern study of free groups is conducted in terms of graphs. So one has the following, rather more specific, precept.
If your group G is free, try to rephrase your question in terms of the topology of graphs.♦
A lot of modern group theory can be seen as an attempt to generalize these techniques to larger classes of groups and spaces: hyperbolic metric spaces and CAT(0) metric spaces, for instance, can be seen in this light.
Here's another example of a fact about free groups that is very simple to prove using topology.
Proof. Let be a free group generated by elements and as above let be a rose with petals, so where is the unique vertex of .
A subgroup of index corresponds to a covering map of degree together with a choice of base vertex in . In particular, is a graph with precisely vertices and edges. Clearly, there are only finitely many such graphs . Furthermore, for each such there are precisely choices of base vertex and only finitely many choices of covering map .
We have seen that can be described by a finite amount of data. This proves the proposition.□
Proposition 3 is particularly useful because, via the universal property of free groups, it follows that the same holds for every finitely generated group.
The Schreier Index Formula is a third nice example.
Suppose is free on generators and is a subgroup of finite index . By Theorem 2 above, is free. But what can we say about the rank of ? There is a very nice answer to this question, using Euler characteristic.
As before, think of as the fundamental group of a rose with petals. The Euler characteristic of is equal to the number of vertices minus the number of edges, so . What's more, Euler characteristic is a homotopy invariant, so it follows that the Euler characteristic of any graph is equal to 1 minus the rank of the fundamental group.
As before, we let be the covering space of corresponding to . We can now compute , the rank of , by double counting. On the one hand, we have seen that . On the other, the Euler characteristic of a covering space is precisely the degree of the covering map multiplied by the Euler characteristic of the base space, so . Equating these and rearranging, we have proved the following theorem.
An example involving trees and amalgamated products, as in Serre's book, would be good here.