Quasi-isometric rigidity of lattices in higher rank simple Lie groups

Taught Course Centre (April-June 2018)

Time: Friday 10am-noon. The TCC timetable can be found here.
Place: Videoconference Room VC1, Mathematical Institute.

Synopsis: Quasi-isometric rigidity results describe properties (most of the time, algebraic properties) that are preserved under the loose metric relation of quasi-isometry of groups endowed with word metrics.

One of the most striking such results is the one stating that a finitely generated group quasi-isometric to SL(n,Z), for n at least 3, is itself SL(n,Z), up to finite index and quotient by a finite normal subgroup.

In this course I shall explain proofs of the theorems of Kleiner-Leeb and of Eskin, extending the result above to lattices (uniform and non-uniform) in higher rank semisimple groups, and their recent improvements due to Fisher-Whyte and Fisher-Nguyen, where quasi-isometries are replaced by quasi-isometric embeddings. I plan to cover the following topics, listed in the order of their appearance in lectures:

Overview of the course. Quasi-isometric rigidity results.

Symmetric spaces of non-compact type: a brief introduction.

Locally symmetric spaces of finite volume. Logarithmic flats.

Spherical and Euclidean buildings.

Ultralimits of symmetric spaces of non-compact type are Euclidean buildings.

The proofs of the Kleiner-Leeb Theorem, and of the Eskin theorem.

If time permits: the recent results of Fisher-Nguyen-Whyte.

The course is based on:

chapters of the book "Geometric Group Theory", written jointly with Misha Kapovich. An older version of the book can be found here.

the paper of Kleiner-Leeb. (IHES Publications); .

my paper proving the Eskin Theorem with methods from Kleiner-Leeb and logarithmic flats; .

the recent preprints of Fisher-Whyte and Fisher-Nguyen proving rigidity for quasi-isometric embeddings, partly based on my methods.

The lecture notes and exercise sheets are posted here:

Due to a technical problem, page 8 has not been saved by the system, therefore I have not combined the odd and even pages of the notes on the whiteboard and am posting them separately:

Lecture 1, odd pages.

Lecture 1, even pages.

Lecture 2.

Lecture 3, odd pages.

Lecture 3, even pages.

Lecture 4, odd pages.

Lecture 4, even pages.

Exercise Sheet 1.

Lecture 5.

Lecture 6.

Lecture 7.

Lecture 8.

Exercise Sheet 2.