# Geometric Analysis and Topology Seminar

#### Zimmer's Conjecture: Subexponential Growth, Measure Rigidity and Strong Property (T)

Speaker: David Fisher, Indiana University

Location: Warren Weaver Hall 805

Date: Tuesday, February 21, 2017, 10 a.m.

Synopsis:

This talk is a sequel to the colloquium of the day before. I will try to make it logically independent and self-contained, but most of the history and motivation will occur in the colloquium talk and this talk will emphasize ideas of proofs of the following theorem. Let $$G$$ be a cocompact lattice in $$\mathrm{SL}(n,\mathbb{R})$$ where $$n>2$$, $$M$$ a compact manifold and $$a: G \rightarrow \mathrm{Diff}(M)$$ a homomorphism. If $$\dim(M) < n-1$$, $$a(G)$$ is finite. Furthermore if $$\dim(M)=n-1$$ and $$a(G)$$ preserves a volume form, $$a(G)$$ is finite. The proof has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result and that it uses results on homogeneous dynamics, including Ratner's measure classification theorem, to prove results about inhomogeneous system. This is joint work with Aaron Brown and Sebastian Hurtado.