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 this course we shall focus primarily on the topological/combinatorial point of view. We shall develop the coarse geometry viewpoint in the Spring semester. We will start with a review of the fundamental group and covering spaces; material that is usually encountered in a first course in topology --- [M] or [H]. We shall study Van Kampen's theorem and some generalizations and the group theory notions of free products with amalgamation and HNN extensions. We shall then follow (loosely) the outline of [SW]; studying Stallings proof of Grushko's theorem, the Kurosh subgroup theorem, a topological treatment of Bass-Serre theory, Stallings theory of groups with infinitely many ends, and Dunwoody's results on accessibility. We may also look at some of the modern developments such as the torus theorem and the JSJ theory as time permits.

- [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. - [H]
- A. Hatcher,
*Algebraic Topology I*, available online (1999) at http://math.cornell.edu/~hatcher. - [M]
- W.S. Massey,
*Algebraic topology: an introduction*, Springer GTM**56**, (1977). - [S]
- J.R. Stallings,
*Group theory and three dimensional manifolds*, Yale Univ. Press, New Haven and London, (1971). - [SW]
- P. Scott and T. Wall,
*Topological methods in group theory*, in ``Homological group theory" (C.T.C. Wall ed.) LMS Lecture Note Series**36**, CUP, (1979).

OU Home Page Math Dept Noel's Home Page Group Theory