Colloquium: Tuesday, April 4, 2 pm. Speaker: Alex Lubotzky (Hebrew University). Title: “First order rigidity of high-rank arithmetic groups”.

The family of high rank arithmetic groups is class of groups which is 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 on them have been proven, including: Mostow rigidity, Margulis super rigidity and the Quasi-isometric rigidity.

We will talk about a new type of rigidity (which at this point we can prove only for many but not all): first order rigidity. Namely if G is such an arithmetic group and H a finitely generated group which is elementary equivalent to it (i.e., the same first order theory in the sense of model theory) 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 whose are low-rank arithmetic groups) are far from having such a rigidity.

Various questions and problems for further research will be discussed.

Joint work (in progress) with Nir Avni and Chen Meiri.