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Use topology to study your group

Quick description

If you want to study a group G, try to realize G as the fundamental group of a topological space. This works best when G is infinite and discrete, especially if G is finitely presented and torsion-free.

Prerequisites

A basic knowledge of group theory and topology, especially the fundamental group and covering spaces.

General discussion

Actions are a good way of studying a group G. The actions of G 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 Xfor brevity, for the remainder of this article such actions shall be referred to as geometric.

If G is the fundamental group of a reasonably nice topological space Y then G acts geometrically on the universal cover, and conversely if G admits such an action on a space X then the quotient space Y has fundamental group G. 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 G by finding a particularly nice topological space X on which G acts, or equivalently by exhibiting G as the fundamental group of a nice space Y. Conversely, attractive hypotheses on Y impose restrictions on G. For instance, if Y is a CW-complex with finite one-skeleton then G is finitely generated (in particular countable), and if Y has finite two-skeleton then G is finitely presented. (This is an equivalent way of looking at group presentations. Any presentation for G describes a two-complex with fundamental group Gthe 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 Y is aspherical then the homology and cohomology of Y are equal to the group homology and cohomology of G, so if Y is also compact then a variety of other conditions are imposed including that G is torsion-free. Therefore, if we want Y to be very nice—compact and aspherical—then G 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 G on X be free. In this case the resulting quotient Y 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 G, try to find a nice space Y with fundamental group G. This is likely to work particularly well if G is finitely presented and torsion-free.

These ideas apply very nicely to free groups.

Example 1

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 n edges is called a rose with n petals. (Here n need not be finite.) The Seifert–van Kampen Theorem implies that the fundamental group of a rose with n petals is precisely the free group on n generators. More generally, let Y be an arbitrary graph and let T be a maximal tree in Y. Then Y/T is a rose and the quotient map Y\to Y/T is a homotopy equivalence. This proves the following.

Theorem 1 A group G is free if and only if G is the fundamental group of a graph.

The Nielsen–Schreier Theorem follows immediately from this and elementary covering-space theory.

Theorem 2 (Nielsen–Schreier Theorem) Every subgroup of a free group is free.

Let F be a non-abelian free group and let H be a subgroup. Let Y be a rose such that F is the fundamental group of Y. By standard covering space theory, there is a covering space Y' of Y with fundamental group H. But a covering space of a graph is a graph, so H 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.

Example 2

Here's another example of a fact about free groups that is very simple to prove using topology.

Proposition 3 A finitely generated free group only has finitely many subgroups of given finite index k.

Proof. Let F be a free group generated by n elements and as above let Y be a rose with n petals, so F\cong \pi_1(Y,v) where v is the unique vertex of Y.

A subgroup H of index k corresponds to a covering map Y'\to Y of degree k together with a choice of base vertex in Y'. In particular, Y' is a graph with precisely k vertices and nk edges. Clearly, there are only finitely many such graphs Y'. Furthermore, for each such Y' there are precisely k choices of base vertex and only finitely many choices of covering map Y'\to Y.

We have seen that H 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.

Example 3

Suppose F is free on n generators and H is a subgroup of finite index k. By Theorem 2 above, H is free. But what can we say about the rank m of H? There is a very nice answer to this question, using Euler characteristic.

As before, think of F as the fundamental group of a rose Y with n petals. The Euler characteristic of Y is equal to the number of vertices minus the number of edges, so \chi(Y)=1-n. 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 Y' be the covering space of Y corresponding to H. We can now compute m, the rank of H, by double counting. On the one hand, we have seen that \chi(Y')=1-m. 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 \chi(Y')=k(1-n). Equating these and rearranging, we have proved the following theorem.

Theorem 4 (Schreier Index Formula) Let F be a free group of rank n and let H be a subgroup of finite index k. Then
 \mathrm{rank}(H)=1-k(1-n).

Example 4

An example involving trees and amalgamated products, as in Serre's book, would be good here.

Comments

This article is very similar

This article is very similar to Actions on topological spaces. Perhaps they should be combined?

Inline comments

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

Also, this is false - it's

Also, this is false - it's homotopy equivalent to a bouquet, but is never a bouquet itself unless it's a homeomorphism. Again, this is already discussed in Actions on topological spaces.

Thanks for catching this

Thanks for catching this — when I wrote this I wasn't mentally distinguishing between homotopy equivalence and homeomorphism.

With regard to your comment on combining pages:

I hadn't looked carefully at Actions on topological spaces; I rather just saw the comment about this result on subgroups of free groups, and thought that it would be a good thing to put into the tricki.

Feel free to combine the articles as you think is best. One thing that might be good would be to follow Tim's encouragement to have imperative titles. (You can see that I followed it in naming this article.) But since you've probably put more work into the actions page than I have into this page, you should do what you think makes the most sense.

Use topology...

I've amalgamated the two articles, and kept the nicer name. A few thoughts that I'd be interested in comments on:

1. Perhaps this is now overloaded with examples about free groups. A new page might be appropriate.

2. A good complement might be a theorem about the fundamental group of a surface. Almost anything would do. One could prove the analogue of the Scheier Index Formula for a closed surface, for instance.

3. Did you have a specific Serre-style example in mind, Matthew? Of course, that stuff is a very nice application of these ideas, and one day I hope to write Use actions on trees to study graphs of groups or whatever, but it seems quite hard to develop in the space available on this page.