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 $\mathrm{SL}(n,\mathbb Z)$, for $n > 2$ , $\mathrm{SL}(n, \mathbb Z[1/p])$ 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)