**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.

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: