First order rigidity of high-rank arithmetic groups

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 is 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.

Based on a joint paper with Nir Avni and Chen Meiri (Invent. 2019)