The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SLn(ℤ), for n ≥ 3, SLn(ℤ[1/p]) for n ≥ 2, their finite-index subgroups and many more. A number of remarkable results about them have been proven including: Weil local rigidity, Mostow strong rigidity, Margulis superrigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. We will add a new type of rigidity: ‘first order rigidity’. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to D then L is isomorphic to D. This stands in contrast with Zlil Sela’s remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non-isomorphic finitely generated groups which are elementarily equivalent to them.
Based on a joint paper with Nir Avni and Chen Meiri.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
