# McGill Geometric Group Theory Seminar

McGill GGT research group

### The seminar takes place on Wednesday at 3 PM in 920 Burnside Hall, unless otherwise noted below.

Seminar organizers: William Chong, Annette Karrer, Zachary Munro, Piotr Przytycki, Daniel Wise

## Upcoming talks:

#### September 28, 2022: Piotr Przytycki (McGill University), Coxeter groups are biautomatic.

Coxeter groups are groups generated by involutions $s_i$, with the relations of form $(s_is_j)^m=id$. For each Coxeter group, we will be discussing a particular system of "voracious" paths between any two vertices of the Cayley graph. This system turns out to have the "fellow traveller" property and is generated by a finite state automaton. This is joint work with Damian Osajda.

#### October 5, 2022: Christopher Karpinski (McGill University), Hyperfiniteness of the Action of Relatively Hyperbolic Groups on their Bowditch Boundary.

An equivalence relation is called hyperfinite if it is an increasing union of equivalence relations with finite equivalence classes. We generalize the work of Marquis and Sabok for the case of hyperbolic groups acting on their Gromov boundary to show that the action of relatively hyperbolic groups on their Bowditch boundary induces a hyperfinite orbit equivalence relation.

## Past talks:

#### September 7, 2022: Annette Karrer (McGill University), The boundary rigidity of lattices in products of trees.

Each complete CAT(0) space has an associated topological space, called visual boundary that coincides with the Gromov boundary in case that the space is hyperbolic. A CAT(0) group $G$ is called boundary rigid if the visual boundaries of all CAT(0) spaces admitting a geometric action by $G$ are homeomorphic. If $G$ is hyperbolic, $G$ is boundary rigid. If $G$ is not hyperbolic, $G$ is not always boundary rigid. The first such example was found by Croke-Kleiner. In this talk we will see that every torsion-free group acting geometrically a product of $n$ regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the boundary homeomorphic to a join of $n$ copies of the Cantor set. This is joint work with Kasia Jankiewicz, Kim Ruane, and Bakul Sathaye.

#### September 14, 2022: Sami Douba (Institut des Hautes Études Scientifiques), A hyperbolic lattice in each dimension with Zariski-dense surface subgroups.

For each integer $n \geq 3$, we exhibit a noncocompact arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.

#### September 21, 2022: Thomas Haettel (Université de Montpellier), Group actions on injective metric spaces.

A metric space is called injective if any pairwise intersecting family of balls has a non-empty global intersection. Such injective metric spaces enjoy many properties typical of nonpositive curvature. In particular, when a group acts by isometries on such a spaces, we can deduce many consequences. We will also present numerous examples of groups acting by isometries on injective metric spaces, including hyperbolic groups, cubulated groups, higher rank lattices, mapping class groups, some Artin groups...