# Columbia Geometric Topology Seminar

Fall 2021

Organizers: Kyle Hayden, Siddhi Krishna
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

## Fall 2021

Date Time (Eastern) Speaker Title

September 17

11 am  (Note nonstandard time)

Proper actions of 3-manifold groups on finite product of quasi-trees

September 24

2 pm

The Khovanov homology of slice disks

October 1

2 pm

Braids, fibered links, and annular Khovanov homology

October 8

2 pm

Geography of spanning surfaces

October 15

2 pm

End-periodic homeomorphisms and volumes of mapping tori

October 22

2 pm

Reeb flows transverse to foliations

November 5

11 am  (Note nonstandard time)

Searching for geometric finiteness using surface group extensions

November 5

Knotted surfaces with infinite cyclic knot group

November 12

11 am (Note nonstandard time)

A reboot of the theory of braids and knots: the Thompson groups as knot constructors

November 19

2 pm

Length, Stable Commutator Length, and Hyperbolic Geometry

December 3

11 am (Note nonstandard time)

Quick quantum invariants of rotational tangles (handout)

December 10

2 pm

Peg Problems

## Abstracts

Wenyuan Yang, Peking University

Title: Proper actions of 3-manifold groups on finite product of quasi-trees

Abstract: Let M be a compact, connected, orientable 3-manifold. In this talk, I will study when the fundamental group of M acts properly on a finite product of quasi-trees. Our main result is that this is so exactly when M does not contain Sol and Nil geometries. In addition, if there is no $\widetilde{SL(2, \mathbb{R})}$ geometry either, then the orbital map is a quasi-isometric embedding of $\pi_1(M)$. This is called property (QT) by Bestvina-Bromberg-Fujiwara, who established it for residually finite hyperbolic groups and mapping class groups. The main step of our proof is to show property (QT) for the classes of Croke-Kleiner admissible groups and of  relatively hyperbolic groups under natural assumptions. Accordingly, this yields that graph 3-manifold and mixed 3-manifold groups have property (QT). This represents joint work with N.T. Nguyen and S.Z. Han.

Isaac Sundberg, Bryn Mawr College

Abstract: A smooth, oriented surface that is properly embedded in the 4-ball can be regarded as a cobordism between the links it bounds, namely, the empty link and its boundary in the 3-sphere. To such link cobordisms, there is an associated linear map between the Khovanov homology groups of the boundary links, and moreover, these maps are invariant, up to sign, under boundary-preserving isotopy of the surface. In this talk, we review these maps and use their invariance to understand the existence and uniqueness of slice disks and other surfaces in the 4-ball. This reflects joint work with Jonah Swann and, separately, with Kyle Hayden.

Gage Martin, Boston College

Abstract: Birman-Hilden give a construction that relates braid closures with certain fibered links via taking a branched double cover. In this talk we will see how the construction can be used to give topological applications of annular Khovanov homology. As an example we will use the Birman-Hilden construction to show that annular Khovanov homology detects a specific 4-braid representative of the unknot. This is joint work with Fraser Binns.

Joshua Howie, UC Davis

Title: Geography of spanning surfaces

Abstract: The geography problem for spanning surfaces asks for a classification of all pairs of slope and euler characteristic which can be realised by a spanning surface for a given knot in the 3-sphere. It is enough to understand the meridionally essential one-sided spanning surfaces, a somewhat larger class of surfaces than the geometrically essential spanning surfaces. We will discuss the existence of such one-sided surfaces, and give an algorithmic solution to the geography problem.

Marissa Loving, Georgia Tech

Title: End-periodic homeomorphisms and volumes of mapping tori

Abstract: I will discuss volumes of mapping tori associated to irreducible end-periodic homeomorphisms of certain infinite-type surfaces, inspired by a theorem of Brock (in the finite-type setting) relating the volume of a mapping torus to the translation distance of its monodromy on the pants graph. This talk represents joint work with Elizabeth Field, Heejoung Kim, and Chris Leininger.

Jonathan Zung, Princeton University

Title: Reeb flows transverse to foliations

Abstract: Eliashberg and Thurston showed that (almost all) C^2 taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.

Anthony ConwayMIT

Title: Knotted surfaces with infinite cyclic knot group.

Abstract: This talk will concern embedded surfaces in 4-manifolds for which the fundamental group of the complement is infinite cyclic. Working in the topological category, necessary and sufficient conditions will be given for two such surfaces to be isotopic. This is based on joint work with Mark Powell.

Jacob RussellRice

Title: Searching for geometric finiteness using surface group extensions

Abstract: Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study and produce surprising applications to  the geometry of surface group extension and surface bundles. In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex  cocompact if  and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged.  Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in MCG(S) might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.

Valeriano AielloUniversität Bern, Mathematisches Institut

Title: A reboot of the theory of braids and knots: the Thompson groups as knot constructors

Abstract: In 2014 Vaughan Jones introduced a method to construct knots from elements of the Thompson groups. More precisely, unoriented knots arise from R. Thompson’s group F, while the oriented ones can be produced out of Jones’s oriented Thompson subgroup. This new framework allows one to start a new theory analogous to that of braids and knots, but with the Thompson groups replacing the braid groups. I will report on some work on this project.

Cameron Rudd, University of Illinois Urbana-Champaign

Title: Length, Stable Commutator Length, and Hyperbolic Geometry

Abstract: Geodesic length and stable commutator length give geometric and topological notions of the complexity for nullhomologous elements of the fundamental group of a hyperbolic manifold. The ratio of these complexity measures is a sort of geometric-topological isoperimetric ratio called the stable isoperimetric ratio. In this talk, I will discuss this ratio and mention how it relates to different aspects of the geometry and topology of hyperbolic manifolds. In particular, I will discuss a connection to the spectrum of the Hodge Laplacian.

Roland van der Veen, University of Groningen

Title: Quick quantum invariants of rotational tangles (handout)

Abstract: We introduce a language of tangles that is sufficiently rich to express interesting properties of knots yet rigid enough to satisfy all the axioms found in quantum groups. Examples of such properties are genus, ribbonness, and being a Whitehead double. Mapping the tangles to tensors in a chosen quantum group in a structure preserving way we get a knot invariant sensitive to the properties we are interested in. In favorable cases these invariants can be calculated efficiently using generating function techniques, making this a useful toolbox for low-dimensional topology. Time permitting we will illustrate our technique with an example related to the (colored) Jones polynomials. This is joint work with Dror Bar-Natan, see our preprint: https://arxiv.org/abs/2109.02057

Joshua Greene, Boston College

Abstract: I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we proved that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.