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.