###### 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.

###### Tuesday, November 6, 2018 at 1:50pm in Math Tower (MW) 154

### Thang Nguyen, New York University

#### Quasi-isometric rigidity of warped cones and expanders

##### Abstract

Warped cone was defined by Roe, and can be used to get an geometric object that carries the information of the action of a group. In the case of actions with spectral gap, we obtain a family of expanders from this construction. We are looking for situations where coarse geometry of warped cone and expanders determines the action. In the talk, I will explain the construction warped cones, expanders together with ideas why we get rigidity results when the spaces that groups acting on are nice enough. Joint work with David Fisher and Wouter van Limbeek.

###### Friday, November 16, 2018 at 4:20pm in Cockins Hall (CH) 232

### Kasia Jankiewicz, University of Chicago

#### Cubical dimension of small cancellation groups

##### Abstract

The cubical dimension of a group `G` is the infimum `n` such
that `G` acts properly on an `n` dimensional `CAT(0)`
cube complex. For each `n` we construct examples of `C’(1/6)`
small cancellation groups with cubical dimension bounded below by `n`.

###### Tuesday, December 4, 2018 at 3pm in Cockins Hall (CH) 240

### Damian Osajda, University of Wrocław and McGill University

#### A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

##### Abstract

This is joint work with Alexandre Martin (Heriot-Watt University). Let
`X` be a complex of hyperbolic groups. In general the fundamental group of
`X` need not to be hyperbolic. M. Bestvina and M. Feighn showed that if
`X` is a graph of groups, and satisfies some natural ‘acylindricity’
conditions then the fundamental group of `X` is hyperbolic. A. Martin
extended this combination theorem to the case of `X` whose underlying
complex carries a hyperbolic `CAT(0)` metric. I will present a
combinatorial counterpart of Martin's result obtained recently. We introduce a weak
nonpositive-curvature-like combinatorial property and show that fundamental groups
of complexes of groups with underlying complex satisfying that property are
hyperbolic. Our property holds for e.g. (weakly) systolic complexes and small
cancellation complexes giving rise to new examples of complexes of groups with
hyperbolic fundamental groups. The proof relies on constructing a potential Gromov
boundary for the resulting groups and analyzing the dynamics of the action on the
boundary in order to use Bowditch’s characterization of hyperbolicity.