Appears in collection : 2018 - T1 - WS1 - Model theory and combinatorics

The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SL(n, Z), for n +/= 2 , SL(n, Z[1/p]) for n +/= 1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity. We will talk about a new type of rigidity : ”first order rigidity”. Namely if G is such a non-uniform characteristic zero arithmetic group and H a finitely generated group which is elementary equivalent to it then H is isomorphic to G. 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 elementary equivalent to them. Joint work with Nir Avni and Chen Meiri.