Geometric Group Theory

Math 828, Spring 2012

Geometric group theory is the study of groups via their actions on metric spaces. Every group is the group of symmetries of some metric space, but this metric space may be hard to deal with. Geometric group theory therefore tends to focus on groups which act on particularly "nice" metric spaces, for instance those satisfying some kind of (coarse or local) curvature bound. In this context curvature bounds tend to be stated in terms of "thin-ness" of triangles. The best possible such bound is satisfied by R-trees: In an R-tree, every triangle is a tripod, and any two points are connected by a unique embedded path. A major part of this course will be devoted to studying actions of groups on R-trees. (In 827 we studied simplicial trees. The results we need from that course will be stated explicitly, so you do not need to have attended 827 to attend this course.) Other topics will be chosen according to the desire of the students and instructor, but will probably include some of:

• Actions of groups on hyperbolic spaces.
• Limits of such actions (these limits are often actions on R-trees).
• The Rips machine for extracting information from an R-tree action.
• JSJ decomposition(s) of a group, which can be thought of as giving structure to the "moduli space" of tree actions of a single group.
• CAT(0) spaces.
• Algorithms to find decompositions of groups.

References

• (Rips machine) M. Bestvina and M. Feighn, "Stable actions of groups on real trees." Inventiones mathematicae 121 (1995), 287-321. MR1346208
• (JSJ decompositions) V. Guirardel and G. Levitt, "JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space", arXiv:0911.3173.
• (CAT(0) spaces) M. Bridson and A. Haefliger, Metric spaces of non-positive curvature.
• (Background on simplicial trees) Jean-Pierre Serre, Trees, MR1954121.
• A more topologically flavored account of amalgams and simplicial tree actions can be found in G.P. Scott and C.T.C Wall, "Topological methods in group theory." MR564422.