##### Department of Mathematics,

University of California San Diego

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### Colloquium

## Aaron Brown

#### The University of Chicago

## Lattice actions and recent progress in the Zimmer program

##### Abstract:

The {\itshape Zimmer Program} is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular, on manifolds whose dimension is below the dimension of all algebraic examples, {\itshape Zimmer's conjecture} asserts that every action is finite. I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my own results within the Zimmer program: (1) a solution to Zimmer's conjecture for actions of cocompact lattices in $SL(n,R), n>=3$ (joint with D. Fisher and S. Hurtado); (2) a classification (up to topological semiconjugacy) of lattice actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).

Hosts: Efim Zelmanov and Amir Mohammadi

### January 9, 2017

### 3:00 PM

### AP&M 6402

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