Brandeis Topology Seminar, Fall 2023

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Ruth Charney (charney@brandeis.edu), Kiyoshi Igusa (igusa@brandeis.edu), Thomas Ng (thomasng@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

September 05: Thomas Ng (Brandeis University)
Title: Improper cubulations of free product quotients

Abstract: Quotients of free products are a rich source of relatively hyperbolic groups with exotic subgroups. Martin and Steenbock introduced a geometric model to show quotients of free products can act properly and cocompactly on CAT(0) cube complexes so long as the free factors did as well. I will discuss joint work with Einstein and Groves that removes the restriction on free factors and yields a special type of improper action of C’(1/6) free products on CAT(0) cube complexes called a relative cubulation.  The main tool is a boundary separation argument that exhibits a sufficiently rich family of quasi-convex subgroups and exploits a characterization of acylindrical hyperbolicity due to Abbott and Manning.   Time permitting, I will also discuss applications to residual finiteness and generic quotients. 

September 12:  Masaki Taniguchi (Kyoto University)
Title：Involutive filtered instanton Floer theory and its applications

Abstract：We present a variant of involutive instanton Floer theory that serves as an obstruction to the existence of diffeomorphisms on certain 4-manifolds with boundaries. Our research yields a new collection of strong corks and strong corks which survive after stabilization by CP^2 or -CP^2. This is joint work with Abhishek Mallick, Irving Dai, and Antonio Alfieri. 

September 19: George Domat (Rice University)

Title: Generating Sets and Coarse Geometry for "Big Out(F_n)"

Abstract: We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina that hopes to answer the question: What is "Big Out(F_n)?" These will arise as groups of proper homotopy classes of proper homotopy equivalences of locally finite graphs. Similar to the surface setting, these groups are not finitely nor compactly generated. As such, one must take more care when attempting to use the standard tools of geometric group theory. We will discuss new results that classify when these groups have a well-defined quasi-isometry type. This is joint work with Hannah Hoganson and Sanghoon Kwak.

September 26: Brandeis Monday (no seminar)

October 03: No seminar

October 10:  Eduard Einstein (Swarthmore College)
Title: Relatively Geometric Actions on CAT(0) Cube Complexes

Abstract: Studying actions of hyperbolic and relatively hyperbolic groups on CAT(0) cube complexes has produced important results in both geometric group theory and low dimensional topology, especially Agol's proof of the Virtual Haken Conjecture. Groves and I introduced relatively geometric actions on CAT(0) cube complexes to study relatively hyperbolic groups that act on CAT(0) cube complexes. In this talk, I will explain how to construct relatively geometric actions using a boundary criterion. I will discuss some of the properties of relatively geometric actions and how we hope to use these tools to develop structure theorems for groups that admit relatively geometric actions. This is primarily joint work with Daniel Groves and may also include joint work with Thomas Ng.

October 17:  Corey Bregman (Tufts University)
Title: Classifying spaces for 3-manifold diffeomorphisms

Abstract: Let M be a compact, connected, orientable 3-manifold, possibly with boundary. The classical Kneser-Milnor theorem states that M admits a decomposition as a connected sum of prime manifolds, unique up to reordering the factors. We parametrize these decompositions on the level of embedded spheres in M, which allows us to build nice models for the classifying spaces BDiff(M) and BDiff(M rel ∂M). When ∂M is nonempty, we use this to show BDiff(M rel ∂M) has finite homotopy type, confirming a conjecture of Kontsevich. Along the way, we discuss the role geometry plays in the structure of 3-manifold diffeomorphisms, and also compute the rational cohomology of BDiff((S^1xS^2)#(S^1xS^2)).  This is joint work with Rachael Boyd and Jan Steinebrunner. 

October 24: Daniel Hartman (University of Georgia)
Title: Smoothing 3-balls in the 5-ball

Abstract: Derived from the work of Freedman and Quinn, it is known that any smooth 2-knot in the 4-sphere with complement having infinite cyclic fundamental group bounds a locally flat 3-ball. However, a significant open question persists: do the 2-spheres bound a smooth 3-ball? An easier question to ask is whether the 2-sphere is smoothly slice, an assertion proven by Kervaire. Consequently, every 3-ball in the 4-sphere is homotopic to a smooth 3-ball in the 5-ball relative to the 2-knot. Between these extremes lies the question of whether the topological 3-ball can be isotoped to a smooth 3-ball in the 5-ball. I will show that this depends entirely on the Rochlin invariant of the 2-knot. Building off of this idea, I will discuss some consequences of the proof as well as show that 2-knots with an inverse under connected sum and that bound a punctured integer homology sphere must have a Rochlin invariant of zero.

October 31:  No seminar

November 07: Hayato Imori (Kyoto University)
Title: Singular instanton homology and knot concordance invariants

Abstract: Floer theory has provided refined invariants to study low-dimensional topology, including knot concordance. In the context of Yang-Mills gauge theory, singular instanton Floer theory gives us numerical knot concordance invariants which refine classical knot signature. In this talk, the speaker will begin by providing a brief overview of the construction of Floer theory from singular instantons. The speaker will also introduce recent developments in the algebraic framework to define concordance invariants. This talk is based on the joint work with Aliakbar Daemi, Kouki Sato, Christopher Scaduto, and Masaki Taniguchi.

November 14: Yandi Wu (University of Wisconsin)
Title: Marked Length Spectrum Rigidity for Surface Amalgams

Abstract: The marked length spectrum of a negatively curved metric space can be thought of as a length assignment to every closed geodesic in the space. A celebrated result by Croke and Otal says that metrics on negatively curved closed surfaces are determined completely by their marked length spectra. Apart from being natural generalizations of surfaces, surface amalgams also have fundamental groups that provide examples of limit groups and finite-index subgroups of RACGs. In my talk, I will discuss my work towards extending Croke and Otal’s result to a large class of surface amalgams.

November 21: No seminar

November 28: Kejia Zhu (UC Riverside)
Title: Relatively geometric actions of complex hyperbolic lattices on CAT(0) cube complexes

Abstract: We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a CAT(0) cube complex, in the sense of \cite{einstein2020relative}. As a consequence, we prove that if $\Gamma$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ that admits a relatively geometric action on a CAT(0) cube complex, then $G$ is isomorphic to $\SO(n,1)$. We also prove that given a relatively hyperbolic group with residually finite parabolic subgroups, if it is K\"ahler and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group. This work is joint with Corey Bregman and Daniel Groves.

December 05:  Elliott Vest (UC Riverside)
Title: Curtain Characterization of Sublinearly Morse Geodesics in CAT(0) Spaces

Abstract: A curtain, developed by Petyt, Spriano, and Zalloum, serves as a combinatorial analogue of a hyperplane for non-cube complex spaces. For a CAT(0) space, this tool enables the construction of an associated hyperbolic space called the curtain model. On the other hand, the sublinearly Morse boundary of a geodesic metric space represents a boundary that captures all the 'hyperbolic directions' of the relevant space. This talk will start with providing background information on both the sublinear Morse boundary and the curtain model of a CAT(0) space. We then show the relationship that the sublinearly Morse boundary of a CAT(0) space continuously injects into the Gromov boundary of its curtain model.