###### Tuesday, August 28, 2018 at 1:50pm in Math Tower (MW) 154

### Mark Pengitore, Ohio State University

#### Unipotent translation-like actions on lattices on lattices in rank 1 simple Lie groups

##### Abstract

The Gersten conjecture says that a group being hyperbolic is equivalent to having
no Baumslag-Solitar subgroups. This is known to be false due to work of Brady. While
there are some weaker versions still open, we are interested in a geometric
reformulation of the Gersten conjecture using translation-like actions. To be more
specific, the geometric Gersten conjecture asks whether hyperbolicity is equivalent
to having no translation-like action by any Baumslag-solitar group. In work in
progress, we show that cocompact lattices in real rank 1 simple Lie groups admit
translation-like actions by cocompact lattices in the unipotent part of the Iwasawa
decomposition of the original Lie group. In particular, we demonstrate that
**Z**` ^{n}` acts translation-like on the fundamental group of any
closed hyperbolic

`n+1`manifold which provides counterexamples to the geometric Gersten conjecture. This is joint work with David Bruce Cohen and Ben McReynolds.

###### Tuesday, September 11, 2018 at 1:50pm in Math Tower (MW) 154

### Jingyin Huang, Ohio State University

#### Uniform lattices acting on RAAG complexes

##### Abstract

It is a classical result by Bieberbach that uniform lattices acting on Euclidean spaces are virtually free abelian. On the other hand, uniform lattices acting on trees are virtually free. This motivates the study of commensurability classification of uniform lattices acting on RAAG complexes, which are cube complexes that "interpolate" between Euclidean spaces and trees. We show the tree times tree obstruction is the only obstruction for commmensurability of label-preserving lattices acting on RAAG complexes.