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

Every finitely generated group G has an associated topological space, called a Morse boundary. It was introduced by a combination of Cordes and Charney--Sultan and captures the hyperbolic-like behavior of G at infinity. In this talk, I will motivate the research on Morse boundaries in several steps. First, I will explain Stalling's theorem -- a fundamental theorem in geometric group theory. Afterward, I will explain an analogous statement for so-called Gromov boundaries of Gromov-hyperbolic groups. As Morse boundaries generalize Gromov boundaries, this raises the question as to whether one can generalize this statement to Morse boundaries. Finally, we will see the relationship between a result on Morse boundaries of graphs of groups and this problem. Results presented are joint with Elia Fioravanti.

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.

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

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...

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.

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.

I will discuss Magnus's Power Series Representation and how to generalize it. This is joint work with Andy Ramirez-Cote.

We will discuss a set of criteria for a subcomplex $Y$ in a compact complex $X$ satisfying $\pi_1$-injectivity. Then, we will discuss under a small-cancellation condition, $\pi_1 X$ is relative hyperbolic to $\pi_1 Y$. The talk is going to be filled with examples and hence serves as a gentle introduction to disk diagrams, small cancellation theory and relative hyperbolicity. This is a joint work with Daniel Wise.

We show that any compact nonpositively curved cube complex Y embeds in a compact nonpositively curved cube complex R where each partial local isometry of Y extends to an automorphism of R. We prove a similar result for compact special cube complexes provided that the partial local isometries satisfy certain conditions. (Joint work with Dani Wise)

We will introduce a class of small cancellation complexes, strict C(6) complexes, and we will prove that they are hyperbolic relative to their maximal $\mathbb Z^2$ subgroups. We will include an introduction to small cancellation and relative hyperbolicity in order to make the talk as accessible as possible. This is joint work with Dani Wise.

Measure equivalence is an equivalence relation on the space
of groups that was defined by Gromov in the 90's as an analytic
analogue of quasi-isometry. Let F be a nonabelian free group. We show
that if $L_1$ and $L_2$ are measure equivalent groups, then the wreath
products $L_1\wr F$ and $L_2\wr F$ are measure equivalent with index
1.

We also make several observations about the way one-ended groups can
live inside a wreath product group $B\wr L$. In particular, we
conclude that if $\phi$ is any automorphism of $B\wr L$ and $L$ is
one-ended, then $\phi(L)$ is conjugate to $L$. This is joint work with
Robin Tucker-Drob.

We are going to do a joint expository talk on the Nielsen-Thurston classification of mapping classes of surfaces and definitions of pretrees, to learn some useful definitions in preparation for Jean-Pierre's talk the following day.

The Nielsen-Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan canonical form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

Dehn surgery is an operation where one constructs a 3-manifold by taking a knot in the 3-sphere, cutting out a tubular neighbourhood and then gluing in another solid torus. Despite its simple nature, there are still many interesting questions about how the topology can be changed under such surgeries. Since the Seifert fibered spaces form an important class of 3-manifolds, it is natural to ask which of these can be obtained by Dehn surgery on a knot in the 3-sphere.

I will begin with a gentle introduction to Dehn surgery and Seifert fibered spaces. If I can successfully convince you that both are interesting, I will use the time remaining to discuss some joint work with Ahmad Issa on the aforementioned question.

Highly influential in the study of mapping class groups was the discovery of Masur--Minsky that the curve graph is hyperbolic, as this provides a natural framework in which to study their negative-curvature properties. From this perspective, curve graphs are in many ways as good as it gets. In this talk I'll discuss a recent analogue of curve graphs that can be defined for any CAT(0) space. Joint with Davide Spriano and Abdul Zalloum.

I will present Garside groups, with basic examples coming from braid groups. I will discuss nonpositive curvature properties of Garside groups, and I will mention recent results concerning Garside structures for some Artin groups.