First order rigidity of high-rank arithmetic groups

Abstract: 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; Weil local rigidity, Mostow strong rigidity, Margulis Super rigidity 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 it 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 elementary equivalent to them.

Joint work with Nir Avni and Chen Meiri (Invent. Math. 217 (219-240) 2019).