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:
Lecture 1, odd pages.
Lecture 1, even pages.
Lecture 3, odd pages.
Lecture 3, even pages.
Lecture 4, odd pages.
Lecture 4, even pages.
Exercise Sheet 1.
Exercise Sheet 2.