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Our first examples of infinite discrete groups will be (non-abelian) free groups. These are most elegantly defined via a universal property.
Definition: Let be a set. A free group on is a group with a set map such that, whenever is a group and is a set map, there exists a unique group homomorphism such that extends .
This says that there’s a correspondence between group homomorphisms and set maps . If you like category theory, you can think of this as the assertion that the functor from to given by is adjoint to the forgetful functor from to .
As usual with objects defined via a universal property, it’s immediate that if 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 is empty then .
Example 2: If then .
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 be a path-connected topological space. Suppose that where and are path-connected open subsets and is also path-connected. For any , the commutative diagram
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 be the rose with petals – that is, the wedge of copies of indexed by . Then .
Proof for finite : The proof is by induction on . In the base case where , we take the wedge of 0 circles to be a point and the proof is immediate.
For the general case, in which is a wedge of circles, let be (a small open neighbourhood of) the circle corresponding to some fixed element and let be (a small open neighbourhood of) the union of the circles corresponding to . Of course , and by induction. Choose an orientation on (ie a choice of direction for each circle) and let be the map that sends to a path that goes round the circle corresponding to .
Consider a set map from to a group . As there is a unique group homomorphism such that . As , there is a unique group homomorphism such that for all . It follows from the Seifert-van Kampen Theorem that there is a unique group homomorphism extending and . QED
We will be interested in finitely generated groups. The free group on is finitely generated if and only if is finite. The cardinality of is called the rank of . 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 be a graph, and let be a maximal tree in . Note that the quotient graph is a rose.
Consider the quotient map . Because is a tree, there is a map , uniquely defined up to homotopy, such that maps each edge of into itself and maps each edge of into itself. It is easy to check that and 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 as the fundamental group of a graph . Let be a subgroup of , and let be the covering space of corresponding to . Then is a graph and so 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 is a subgroup of of finite index then the rank of is .
Proof: Again, let and let , where is the rose with petals and is a covering space of . It is standard that
If is the rose with petals then it’s clear that . Similarly, the rank of is . This completes the proof. QED
- Prove that has a subgroup isomorphic to , for any countable .
- Prove that has a subgroup of infinite index isomorphic to , for any countable .
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.
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.
- 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, (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.