On Zimmer's conjecture

On Zimmer's conjecture

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Sebastian Hurtado , University of Chicago
IAS Room S-101

The group $Sl_n({\mathbb Z})$ (when $n > 2$) is very rigid. For example, Margulis proved all its linear representations come from representations of $Sl_n({\mathbb R})$ and are as simple as one can imagine. Zimmer's conjecture states that certain "non-linear" representations (group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions. For example, conjecturally the only action on $Sl_n({\mathbb Z})$ on an $(n-1)$ dimensional manifold (up to some trivialities) is the one on the $(n-1)$ sphere coming projectivizing natural action of $Sl_n({\mathbb R})$ on ${\mathbb R}^n$. I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.