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1. Free groups

21 January 2009 in Course notes | Tags: Free groups | by Henry Wilton | 2 comments

Our first examples of infinite discrete groups will be (non-abelian) free groups. These are most elegantly defined via a universal property.

Definition: Let $S$ be a set. A free group on $S$ is a group $F_S$ with a set map $i:S\to F_S$ such that, whenever $G$ is a group and $f:S\to G$ is a set map, there exists a unique group homomorphism $\hat{f}:F_S\to G$ such that $\hat{f}\circ i$ extends $f$ .

This says that there’s a correspondence between group homomorphisms $F_S\to G$ and set maps $S\to G$ . If you like category theory, you can think of this as the assertion that the functor from $\mathrm{Set}$ to $\mathrm {Grp}$ given by $S\mapsto F_S$ is adjoint to the forgetful functor from $\mathrm{Grp}$ to $\mathrm{Set}$ .

As usual with objects defined via a universal property, it’s immediate that if $F_S$ exists then it’s unique up to unique isomorphism. If we were to embrace the categorical point of view fully, then we could guarantee the existence of free groups by appealing to an adjoint functor theorem.

Example 1: If $S$ is empty then $F_S\cong 1$ .

Example 2: If $S=\{s\}$ then $F_S\cong\mathbb{Z}$ .

To construct free groups on larger sets, we adopt a topological point of view. Our main tool will be the Seifert-van Kampen Theorem.

Seifert-van Kampen Theorem: Let $X$ be a path-connected topological space. Suppose that $X=U\cup V$ where $U$ and $V$ are path-connected open subsets and $U\cap V$ is also path-connected. For any $x\in U\cap V$ , the commutative diagram

eqlatex

is a push out.

There are more sophisticated versions of this, which enable one to think about arbitrarily large open covers. But this will be sufficient for most of our purposes.

Theorem 1: Let $X$ be the rose with $|S|$ petals – that is, the wedge of $|S|$ copies of $S^1$ indexed by $S$ . Then $\pi_1(X)\cong F_S$ .

Proof for finite $S$ : The proof is by induction on $|S|$ . In the base case where $S=\varnothing$ , we take the wedge of 0 circles to be a point and the proof is immediate.

For the general case, in which $X$ is a wedge of $|S|$ circles, let $U$ be (a small open neighbourhood of) the circle corresponding to some fixed element $s_0\in S$ and let $V$ be (a small open neighbourhood of) the union of the circles corresponding to $T=S\smallsetminus{s_0}$ . Of course $\pi_1(U)\cong\mathbb{Z}$ , and $\pi_1(V)\cong F_T$ by induction. Choose an orientation on $X$ (ie a choice of direction for each circle) and let $i:S\to\pi_1(X)$ be the map that sends $s$ to a path that goes round the circle corresponding to $s$ .

Consider a set map $f$ from $S$ to a group $G$ . As $\pi_1(U)\cong\langle i(s_0)\rangle$ there is a unique group homomorphism $f_1:\pi_1(U)\to G$ such that $f_1\circ i(s_0)= f(s_0)$ . As $\pi_1(V)\cong F_T$ , there is a unique group homomorphism $f_2:\pi_1(V)\to G$ such that $f_1\circ i(t)= f(t)$ for all $t\in T$ . It follows from the Seifert-van Kampen Theorem that there is a unique group homomorphism $\hat{f}:\pi_1(X)\to G$ extending $f_1$ and $f_2$ . QED

We will be interested in finitely generated groups. The free group on $S$ is finitely generated if and only if $S$ is finite. The cardinality of $S$ is called the rank of $F_S$ . We will see a bit later that this is an invariant of the isomorphism type of a free group.

Theorem 1 implies that every free group is the fundamental group of a graph (ie a one-dimensional CW-complex). This has a strong converse.

Theorem 2: A group is free if and only if it is the fundamental group of a graph.

The idea of the proof is fairly straightforward: simply contract a maximal tree.

Proof: By Theorem 1, it is enough to show that every graph is homotopy equivalent to a rose. Let $G$ be a graph, and let $T$ be a maximal tree in $G$ . Note that the quotient graph $G'=G/T$ is a rose.

Consider the quotient map $q: G\to G'$ . Because $T$ is a tree, there is a map $r:G'\to G$ , uniquely defined up to homotopy, such that $q\circ r$ maps each edge of $G'$ into itself and $r\circ q$ maps each edge of $G\smallsetminus T$ into itself. It is easy to check that $q\circ r$ and $r\circ q$ are homotopic to the identity maps. QED

We are now in position to give a very easy topological proof of a fairly sophisticated group-theoretic result.

Nielsen-Schreier Theorem: Every subgroup of a free group is free.

Proof: Think of a free group $F$ as the fundamental group of a graph $X$ . Let $H$ be a subgroup of $F$ , and let $X'$ be the covering space of $X$ corresponding to $H$ . Then $X'$ is a graph and $H=\pi_1(X')$ so $H$ is free. QED

Another theorem that we can now easily prove relates the rank of a finite-index subgroup of a free group to its index.

Schreier Index Formula: If $H$ is a subgroup of $F_r$ of finite index $k$ then the rank of $H$ is $1+k(r-1)$ .

Proof: Again, let $F_r=\pi_1(X)$ and let $H=\pi_1(X')$ , where $X$ is the rose with $r$ petals and $X'$ is a covering space of $X$ . It is standard that

$\chi(X')=k\chi(X)$ .

If $X$ is the rose with $r$ petals then it’s clear that $\chi(X)=1-r$ . Similarly, the rank of $H$ is $1-\chi(X')$ . This completes the proof. QED

Exercise 1:

Prove that $F_2$ has a subgroup isomorphic to $F_k$ , for any countable $k$ .
Prove that $F_2$ has a subgroup of infinite index isomorphic to $F_k$ , for any countable $k$ .

Syllabus

1 January 2009 in Syllabus | by Henry Wilton | Leave a comment

The approximate syllabus for the course, roughly as available on the departmental website.

INTRODUCTION

Geometric group theorists study infinite groups via their actions on geometric spaces and, conversely, study geometric spaces via the groups that act on them. This course aims to explore some ideas surrounding the following question.

Motivating Question: “Is every word-hyperbolic group residually finite?”

To give an idea of how wide-ranging the implications of this question are, a positive answer would imply that every compact manifold with negative sectional curvature bounded away from 0 has a non-trivial finite covering space.

Two important themes covered in this course will be:-

1. Word-hyperbolic groups – finitely presented groups that exhibit a coarse form of negative curvature.

2. Residual finiteness and its generalizations. A group is residually finite if no element dies in every finite quotient. In particular, an infinite residually finite group has a lot of finite quotients.

PREREQUISITES

Students in this course will be expected to have seen the basic notions of algebraic topology, specifically: the fundamental group, covering spaces, van Kampen’s Theorem, and homology. Some knowledge of the geometry of the hyperbolic plane would also be useful.

I FREE GROUPS

We will start with the motivating example of free groups, which we can think of as precisely the fundamental groups of graphs. Many classical theorems that seemed quite hard when proved using the techniques of combinatorial group theory are quite easy when viewed from this point of view.

II RESIDUAL FINITENESS AND ITS GENERALIZATIONS

A particular example of a theorem about free groups that is quite easy from a topological point of view is Marshal Hall’s Theorem, which is a very strong generalization of the fact that free groups are residually finite. Stallings gave a beautiful topological proof of this result. In this chapter we’ll also talk about some other topics related to residual finiteness – the profinite topology, separability, the topological interpretation of separability, and Selberg’s Lemma. We’ll also talk a little about the relevance of separability to the study of 3-manifolds.

III WORD-HYPERBOLIC GROUPS

A group is word-hyperbolic if its Cayley graph satisfies a certain coarse negative curvature condition. Examples include free groups and the fundamental groups of hyperbolic manifolds. In this section we will learn about various basic notions in geometric group theory, including quasi-isometries and the Svarc–Milnor Lemma. We will also talk about some deep unresolved open questions about word-hyperbolic groups.

IV BASS–SERRE THEORY (ACCORDING TO SCOTT AND WALL)

Bestvina and Feighn’s combination theorem gives a powerful way of constructing many examples of hyperbolic groups. To study them, we need to understand Bass–Serre theory, which basically asserts that splittings of groups are the same as group actions on trees. Fortunately, this is easy to understand from the point of view of Scott and Wall, who interpret Bass–Serre theory in terms of graphs of spaces.

V EXTENDING STALLINGS’ IDEAS

In this chapter, we’ll prove that a double of two free groups along a maximal cyclic subgroup is subgroup separable. This result extends the ideas of Stallings from the setting of graphs to graphs of spaces. In order to do this, we’ll need to make a detailed study of the theory of covering spaces of graphs of spaces. The main tools will be elevations, which describe the edge maps in coverings of graphs of spaces.

If there’s time, it would be nice to cover some of the following.

VI COMBINATORIAL DEHN FILLING

Combinatorial Dehn filling, due to Groves–Manning and Osin independently, generalizes the usual notion of Dehn filling along a boundary component of a hyperbolic 3-manifold to arbitrary relatively hyperbolic groups. In this chapter, we’ll learn a little about the definition of relatively hyperbolic groups, and then go on to understand the statement of the Groves–Manning–Osin Theorem.

VII THE IMPLICATIONS OF COMBINATORIAL DEHN FILLING

Combinatorial Dehn Filling has raised the stakes on the question that motivates this course. A positive answer would imply some very strong facts about the residual properties of word-hyperbolic groups, including that all closed hyperbolic 3-manifolds are subgroup separable. In this chapter, we’ll talk about theorems of Agol–Groves–Manning and Delzant–Gromov–Olshanksii, which establish these implications.

VIII CONJUGACY SEPARABILITY

Another consequence of a positive answer to the Motivating Question would be that many word-hyperbolic groups would be conjugacy separable. If we have time, we’ll talk about a way of interpreting this geometrically, and give a proof that free groups are conjugacy separable.

BACKGROUND READING

M. Gromov, Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987
Jean-Pierre Serre, Arbres, amalgames, $\mathrm{SL}_2$ (French), Rédigé avec la collaboration de Hyman Bass. Astérisque, No. 46. Société Mathématique de France, Paris, 1977. 189 pp.
Peter Scott and Terry Wall, Topological methods in group theory. Homological group theory (Proc. Sympos., Durham, 1977), pp. 137–203, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979.
Pierre de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. vi+310 pp.

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