Bass-Serre Theory and Complexes of Groups

This course is about groups acting on CW-complexes. We will start with the important case where the CW-complex is one-dimensional. This is usually called Bass-Serre Theory, and gives various generalizations of the Seifert van-Kampen Theorem in topology. The Seifert van-Kampen Theorem expresses the fundamental group of a space as a graph of groups -- in that case with two vertex groups and a single edge group.

We will also talk about Serre's "Property FA". If a group has Property FA, it cannot be analyzed using actions on trees, so we have to look to actions on higher-dimensional complexes. (An example is the building associated to SL(n,F), where F is a field with a discrete valuation.)

This is where the theory of complexes of groups comes in. This theory is more powerful in the presence of some kind of non-positive curvature condition, so we will also talk a bit about CAT(0) spaces.

Related to the theory of complexes of groups is that of orbifolds (generalizations of manifolds which include some "local group" information analogous to the vertex groups of a graph of groups), and if time permits we will put all these ideas in the appropriate common context, which is locally convex groupoids.

Here are some readings. Links should work on campus, or off-campus, using PassKey.

Bass-Serre Theory (actions on trees)

P. Scott and T. 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 MR0564422.
Serre, Trees. MR1954121.

Orbifolds

[CHK] D. Cooper, C.D. Hodgson, S.P. Kerckhoff Three-dimensional Orbifolds and Cone-Manifolds.

Complexes of Groups

[BH] M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, especially chapters IIII.C and III.G.
[GM] D. Groves and J.F. Manning, "Hyperbolic groups acting improperly," arXiv:1808.02325.

Here are some extremely unpolished notes (Updated 2022-09-14, future versions will be on Overleaf -- ask me for an invite to the project if you don't have one) on some of what we've covered so far.

Jason Manning's home page.

Last Updated 2022-08-25