Seminar organizers: William Chong, Katherine Goldman, Christopher Karpinski, Carl Kristof-Tessier, Piotr Przytycki, Daniel Wise
In 2022, Osajda and Przytycki showed that any Coxeter group $W$ is biautomatic. Key to their proof is the notion of voracious projection of an element $g \in W$, which is used iteratively to construct a biautomatic structure for $W$: the voracious language. In this talk, I will generalize these two notions by defining them for any Garside shadow $B$ in a Coxeter system $(W,S)$. This leads to the result that any finite Garside shadow in $(W,S)$ can be used to construct a biautomatic structure for $W$. In particular, for the Garside shadow $L$ of low elements, the biautomatic structure obtained corresponds to the original voracious language of Osajda and Przytycki. These results answer a question of Hohlweg and Parkinson.
A hyperbolic group $G$ is said to have globally stable cylinders if there exist integers $K, m$ and a collection of equivariant $(1,K)$-quasi-geodesics connecting every pair of points in $G$ such that the following holds. For any $x,y,z\in G$; the (aforementioned) quasi-geodesics connecting them form a tripod, up to removing at most $m$ balls of radius $K$ centered along such quasi-geodesics. In short, the property asks for the existence of a bicombing on $G$ satisfying some very strong properties making the group look as close to a tree as possible. In 1995, Rips and Sela asked if torsion-free hyperbolic groups have this property, and in 2022 Sageev and Lazarovich established it for cubulated hyperbolic groups (hyperbolic groups with a geometric action on a CAT(0) cube complex). I will discuss recent work with Petyt and Spriano showing that residually finite hyperbolic groups admit globally stable cylinders.
A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection of Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite.
I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space $\text{BDiff}(M)$, for M a compact, connected, reducible 3-manifold. We prove that when $M$ is orientable and has non-empty boundary, $\text{BDiff}(M\text{ rel } ∂M)$ has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where $M$ is irreducible by Hatcher and McCullough. The theory we develop to prove this theorem has other applications, and I'll provide an overview of these.