# Math 6843     Geometric Group Theory II         Spring 2000

## Announcement

Geometric group theory has its origins in the work of Max Dehn [D] who first formulated the word and isomorphism problems for infinite groups and provided a beautiful solution to the word problem for the fundamental groups of closed hyperbolic surfaces.

There have been two main influences this century which have shaped modern geometric group theory. The first influence came from the combinatorial and topological techniques of 3-manifold theory, a lot of which were developed and carried over by John Stallings [S]. The second major influence, pioneered by Mikhael Gromov [G1], [G2] and [G3], involves studying infinite groups as geometric objects. One of Gromov's key contributions in this area is the development of the concept of large scale (or coarse) hyperbolicity, and the realization that the large scale phenomena associated with negative curvature have some dramatic consequences in group theory.

In the fall course we focused primarily on the topological/combinatorial point of view. This semester we shall develop the (coarse) geometry viewpoint. The main theme of this course will be nonpositive curvature in group theory. We shall begin by studying the word problem in free groups, and then in the fundamental groups of closed hyperbolic surfaces (following Max Dehn). Then we shall study Gromov's hyperbolic groups and see that they too have a Dehn's algorithm solution to the word problem. On many occasions in the course, we will need to construct explicit examples of groups. We indroduce the notions of CAT(0) and CAT(-1) spaces, and piecewise euclidean/hyperbolic complexes for this purpose. We shall study many interesting families of CAT(0) and CAT(-1) groups and their subgroups.

The prerequisites for this course include a working knowledge of fundamental groups and covering spaces, and an interest in combinatorial geometry. If you are intrigued by this material but have not taken the fall course, you should talk with me to determine the precise background requirements.

## References

[D]
M. Dehn, Papers on group theory and topology, translated and introduced by J. Stillwell, Springer New York (1987).
[G1]
M. Gromov, Infinite groups as geometric objects, Proceedings of the I.C.M. Warsaw 1, (1984), 385--392.
[G2]
M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53 (1981), 53--73.
[G3]
M. Gromov, Hyperbolic Groups, in ``Essays in Group Theory", S.M. Gersten (ed.), M.S.R.I. Publ. 8 (Springer 1987), 75--263.
[S]
J.R. Stallings, Group theory and three dimensional manifolds, Yale Univ. Press, New Haven and London, (1971).