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Subexponential growth, measure rigidity, strong property (T) and Zimmer's conjecture

Presented by: 
David Fisher
Thursday 11th May 2017 - 16:00 to 17:00
INI Seminar Room 1
Co-authors: Aaron Brown (University of Chicago), Sebastian Hurtado (University of Chicago)

Lattices in higher rank simple Lie groups, like SL(n,R) for n>2, are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds. After providing some history and motivation, I will discuss a very recent result, proving many cases of the main conjecture. The proof has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result, that it uses results on homogeneous dynamics, including Ratner's measure classification theorem, to prove results about inhomogeneous system and that it uses analytic notions originally defined for the purposes of studying the K theory of C^* algebras. 
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons