# Columbia Geometric Topology Seminar

Fall 2020

Organizer: Nick Salter.
The GT seminar typically meets on Fridays at 2:00pm Eastern time via the Zoom link above. (The password is `math').

Other area seminars. Our e-mail list. Archive of previous semesters

## Spring 2020

Date Speaker Title
September 25
October 2
October 9
October 16 Yulan Qing Sublinearly Morse boundary of groups
October 23 (11 AM) Federica Fanoni Homeomorphic subsurfaces and omnipresent arcs
October 30 Asaf Hadari  Mapping class groups which do not virtually surject to the integers.
November 6 Katie Mann Stability for hyperbolic groups acting on their boundaries
November 13 Saul Schleimer Cohomology fractals
November 20 (11 AM) Anthony Genevois How to braid a group
December 4 (11 AM) Arnaud Mortier Cohomology of the space of long knots
December 11 Dan Margalit Homomorphisms of braid groups
December 18

## Abstracts

Dan Margalit, Georgia Tech

In the early 1980s Dyer-Grossman proved that every automorphism of the braid group is geometric, meaning that it is induced by a homeomorphism of the corresponding punctured disk. I'll discuss two recent generalizations. With Lei Chen and Kevin Kordek, we prove that every homomorphism from the braid group on n strands to the braid group on (up to) 2n strands is geometric.  With Kordek, we prove that every homomorphism from the commutator subgroup of the braid group to the braid group is geometric. Both results can be interpreted in terms of maps between spaces of polynomials.  We will begin with some background, explain the statements of both theorems, and discuss the basic ideas behind both of the proofs.  We'll focus in particular on a new tool called totally symmetric sets.

Arnaud Mortier, Caen

In the early 1990s, Victor A. Vassiliev developed a powerful approach to knot invariants by considering the set of knots as a stratified topological space, yielding a spectral sequence that computes cohomology classes "of finite complexity". Joan Birman and Xiao-Song Lin then rewrote Vassiliev’s equations to expose the crucial role played by what is now known as the 4T relation, which allowed later on for several universal Vassiliev invariants to be found (0-cohomology classes that contain as much information as they possibly could). Yet, little was done to investigate higher degree invariants. I will introduce in this talk a construction that could turn out to be a universal 1-cohomology class, in the spirit of the Kontsevich integral.

Anthony Genevois, Universite Paris

November 20, 2020
Title
How to braid a group
Abstract:
In this talk, I will describe asymptotically rigid mapping class groups of some surfaces and explain how they can be used to construct "braided" versions of classical groups, including Thompson's and Houghton's groups. Next, I will explain how to make our asymptotically rigid mapping class groups act on contractible cube complexes with stabilisers isomorphic to finite extensions of braid groups. The rest of the talk will be dedicated to various applications of this construction, including proofs of Funar-Kapoudjian's and Degenhardt's conjectures regarding finiteness properties of braided Ptolemy-Thompson's and Houghton's groups. (Joint work with A. Lonjou and C. Urech.)

Saul Schleimer, University of Warwick

We introduce cohomology fractals; these are a sort of dual to Cannon-Thurston maps for hyperbolic three-manifolds.  We have implemented these using a novel ray-tracing algorithm.  It is designed to be fast; in fact, it runs in your browser!  After recalling some of the history of hand- and computer-drawn images of Kleinian groups, I will explain both cohomology fractals and the implementation of the browser application.This is joint work with David BachmanMatthias Goerner, and Henry Segerman.  Please see our YouTube videos, our expository paper, or our research paper for more details.

Katie Mann, Cornell University

November 6, 2020
Title
Stability for hyperbolic groups acting on their boundaries
Abstract:
A hyperbolic group acts naturally by homeomorphisms on its boundary.  The theme of this talk is to say that, in many cases, such an action has very robust dynamics.

Jonathan Bowden and I studied a very special case of this, showing if G is the fundamental group of a compact, negatively curved Riemannian manifold, then the action of G on its boundary is topologically stable (small perturbations of it are semi-conjugate, containing all the dynamical information of the original action).   In new work-in-progress with Jason Manning, we get rid of the Riemannian geometry and show that such a result holds for (hopefully all) hyperbolic groups with sphere boundary, using purely large-scale geometric techniques.

This theme of studying topological dynamics of boundary actions dates back at least as far as work of Sullivan in the 1980's, although we take a very different approach.  My talk will give some history and some picture of the large-scale geometry involved in our work.

October 30, 2020
Title: Mapping class groups which do not virtually surject to the integers
Abstract:
Mapping class groups of surfaces of genus at least 3 are perfect - their only abelian quotient is the trivial group. They do however have finite index subgroups with non-trivial abelianizations. A long standing conjecture of Ivanov states that in sufficiently high genus, the abelianization of a finite index subgroup of the mapping class group is always finite. We will discuss our recent proof of this conjecture, which uses a theorem of Putman and Wieland relating Ivanov's conjecture to a representation theoretic criterion.

Federica Fanoni, CNRS

Homeomorphic subsurfaces and omnipresent arcs
Abstract:
In this talk I will discuss how we can see infinite-type surfaces (e.g. surfaces of infinite genus) as subsurfaces of themselves in a nontrivial way. I will show how to use this fact to select a class of (proper) arcs on a surface and to construct a graph with an interesting mapping class group action. Joint work with Tyrone Ghaswala and Alan McLeay.

Yulan Qing, Fudan University

October 16, 2020
Title
Sublinearly Morse boundary of groups
Abstract:
Gromov boundary plays a central role in many aspects of geometric group theory. In this study, we develop a theory of boundary when the condition on hyperbolicity is removed: For a given proper, geodesic metric space X and a given sublinear function $\kappa$, we define the $\kappa$-boundary, as the space of all sublinearly-Morse quasi-geodesics rays. The sublinearly Morse boundary is QI-invariant and thus can be associated with the group that acts geometrically on X. For a large class of groups, we show that sublinearly Morse boundaries are large: they provide topological models for the Poisson boundaries of the group. This talk is based on several joint projects with Ilya Gekhtman, Kasra Rafi and Giulio Tiozzo.