Fixed point properties and proper actions on non-positively curved spaces and on Banach spaces

Taught Course Centre (April-June 2016)

Time: Friday 11am-1pm. The TCC timetable can be found here.
Place: Videoconference Room VC1, Mathematical Institute.

Synopsis: One way of understanding groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc.

Classical properties such as Kazhdan's property (T) and the Haagerup property are formulated in terms of such actions and turn out to be relevant in a wide range of areas, e.g. combinatorics, algebra, dynamics and in relation to the conjectures of Novikov and Baum-Connes.

In this course I shall overview property (T) and Haagerup, and their various generalisations to Banach spaces, especially in connection with classes of groups acting on non-positively curved spaces.

The course is mainly based on chapters of the book "Geometric Group Theory", written jointly with Misha Kapovich. The file of the book is available here. Comments and corrections are most welcome.

Lecture notes:

Lecture 1.

Lecture 2.

Exercise Sheet 1.

Lecture 3.

Lecture 4.

Lecture 5.

Exercise Sheet 2.

Lecture 6.

Lecture 7.

Exercise Sheet 3 (corrected).

Lecture 8.