# Geometric Structures on 3-manifolds

## Proper affine actions of right angled Coxeter groups

Jeffrey Danciger
University of Texas, Austin
March 8, 2016
We prove that any right-angled Coxeter group on $k$ generators admits a proper action by affine transformations on $\mathbb R^{k(k-1)/2}$. As a corollary, many interesting groups admit proper affine actions including surface groups, hyperbolic three-manifold groups, and Gromov hyperbolic groups of arbitrarily large virtual cohomological dimension. Joint work with Francois Gueritaud and Fanny Kassel.

## Morse index and multiplicity of min-max minimal hypersurfaces

Fernando Codá Marques
Princeton University
March 1, 2016
The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. Nothing was said also about the fundamental problem of multiplicity. In this talk I will describe our current efforts to develop the theory further. I will discuss the first general Morse index bounds for minimal hypersurfaces produced by the theory. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.