Schedule:

We are meeting in BA6180 from September 10 to December 3.

About:

  1. I will try to cover the backbone from chapters 2 to 6 of the main book. If possible I will talk about chapter 7, but I am not sure we will be able to talk too much about.

Bibliography:

[1] Buildings. Peter Abramenko, Kenneth S. Brown. Springer, Graduate Texts in Mathematics, Vol. 248

Logbook:

DateWhat did we do?Notes
September 10We talked about properties of pairs (W, S).
We defined the deletion, exchange and folding
conditions and discussed parabolic subgroups.
[1] Sections 2.3.1 and 2.3.2.
See also [1] section 1.5
N/A
September 17We introduced the action and the coxeter conditions and defined Coxeter Group. We defined what the associated Coxeter Complex is and saw an example of it.
[1] Section 2.1, 2.3.3, 2.4, 3.1
See section 2.2 to see examples.
PDF
September 24We finished the topics in the notes of previous time: we proved the properties of the Coxeter Complex, [1] Theorem 3.5. We also defined the components associated to an automorphism and discussed why they satisfy the conditions they do: [1] Lemma 3.31.
[1] Section 3.1, 3.3.1

Our discussion about the reason why the complex is thin was confusing. Here is a written explicit proof: THIN.
PDF
October 1PDF
October 8PDF
October 15PDF
October 22 PDF
October 29 PDF
November 5 PDF
November 12 PDF
November 19 PDF
November 26 PDF
December 3 PDF

Notes:

  • The book [1], for U of T students, can be obtained as a MyCopy book in springer through the UofT libraries.
  • Chapter 1 of [1] covers finite reflection groups, which are the objects we are trying to generalize to build coxeter systems. From them we will build buildings. It is very instructive to review this chapter as we progress and see how it is done in the case of finite W.

Alternative References:

#Reference:
1An application of the bulding to orbital integrals
Jonathan D. Rogawski
2Twin Buildings and Application to S-Arithmetic Groups
Peter Abramenko
3Buildings and their applications in Geometry and Topology
Lizhen Ji