Spring 2022
Meeting dates: Tuesdays (Topology) 1:50-2:50 pm and Thursdays (Geometric Group Theory) from 10:20 to 11:20 am ET
Seminar organizers - Jingyin Huang, Heejoung Kim, Christoforos Neofytidis, and Rachel Skipper
Spring 2022 Schedule
The seminars will be run partially in person and partially virtually over Zoom for the Spring 2022 Semester
https://osu.zoom.us/j/93661626526?pwd=b2xiSEJSTm9BRVlRSitOZXVkMVMzZz09
Zoom ID: 936 6162 6526, Password: 273789
Jan 13-Bin Sun (University of Oxford)
Wreath-like product groups and rigidity of their von Neumann algebras
We construct the first example to the Connes’ Rigidity Conjecture and one of Jones Millennium Problems. More specifically, for every finitely presented group Q, we construct an infinite ICC property (T) group G such that (i) if a group H has the same group von Neumann algebra as G, then H ∼= G and (ii) the outer automorphism group of the group von Neumann algebra of G is isomorphic to Q. Our construction uses a generalization of wreath product, which we call wreath-like product. This is a joint work with Ionut Chifan, Adrian Ioana and Denis Osin.
Jan 18- Dawid Kielak (Oxford)
Twisting L²-Betti number
We will explain why it is important from the computational point of view to be able to twist L²-Betti numbers by finite dimensional representations. A specific behaviour of these Betti numbers has been conjectured by Lück. The conjecture has now been confirmed for locally indicable and sofic groups, and we will discuss these new developments.
Jan 25- Stefan Stadler (Max-Planck Institute of Mathematics)
CAT(0) manifolds
In the early eighties, Gromov asked if a topological manifold which supports a metric of non-positive curvaturein a synthetic sense has to be homeomorphic to the Euclidean space.In the talk, I will discuss Gromov's question and report on the following result obtained jointly with Alexander Lytchak and Koichi Nagano: 4-dimensional CAT(0) manifolds are homeomorphic to the Euclidean space.
Jan 27- Caterina Campagnolo (Madrid)
A Gromov norm-mass inequality for complete manifolds
Since Gromov defined the simplicial volume in 1982 and discovered its relation with the minimal volume, it has been a recurrent theme in the field to obtain constraints and relations between volume and simplicial volume of a manifold.
In joint work with Shi Wang, we extend an inequality of Besson-Courtois-Gallot about the Gromov norm of the fundamental class of a compact manifold to all homology classes of a complete manifold. The inequality is sharper than Gromov's original one and is expressed in terms of the critical exponent of the fundamental group of the manifold.
I will define all necessary objects, give some context and the main proof ideas for the result.
Feb 1-Elizabeth Field (University of Utah)
End periodic homeomorphisms and volumes of mapping tori
In this talk, we will discuss mapping tori associated to irreducible, end periodic homeomorphisms of certain infinite-type surfaces. Inspired by a theorem of Brock in the finite-type setting, we will relate the minimal convex core volume of such a mapping torus to the translation distance of its monodromy on (a certain subgraph of) the pants
graph. This talk represents joint work with Heejoung Kim, Christopher Leininger, and Marissa Loving.
Feb 3- Harry Petyt (University of Bristol)
Hyperbolic models for CAT(0) spaces
CAT(0) spaces are a classical family of nonpositively curved spaces. Among these, CAT(0) cube complexes have received special attention because they have a combinatorial structure that makes them especially nice to work with. One use of this structure has been the construction of hyperbolic spaces associated with CAT(0) cube complexes that "detect their hyperbolic parts". In this talk, we shall discuss ongoing work with Davide Spriano and Abdul Zalloum, in which we give a new approach to constructing hyperbolic spaces that does not rely on the combinatorial structure, and hence applies to all CAT(0) spaces. This has applications to rank-one isometries.
Feb 10- Marco Moraschini (Bologna)
Aspherical manifolds and simplicial volume
Simplicial volume is a homotopy invariant for compact manifolds introduced by Gromov in the early 80s. It measures the complexity of a manifold in terms of singular simplices. Since simplicial volume behaves similarly to the Euler characteristic, a natural problem is to understand the relation between these two invariants. More precisely, a celebrated question by Gromov (~’90) asks whether all oriented closed connected aspherical manifolds with zero simplicial volume also have vanishing Euler characteristic. In this talk, we will describe the problem and we will show counterexamples to some variations of the previous question. Moreover, we will describe some new strategies to approach the problem as well as the relation between Gromov’s question and other classical problems in topology. This is part of a joint work with Clara Löh and George Raptis.
Mar 1- Jingyin Huang (OSU)
Morse quasiflats
We are motivated by looking for traces of hyperbolicity in a space or group which is not Gromov-hyperbolic. One previous approach in this direction is the notion of Morse quasigeodesics, which describes "negatively-curved" directions in the spaces; another previous approach is "higher rank hyperbolicity" with one example being that though triangles in products of two hyperbolic planes are not thin, tetrahedrons made of minimal surfaces are "thin". We introduce the notion of Morse quasiflats, which unifies these two seemingly different approaches and applies to a wider range of objects. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and comment on potential applications. Based on joint work with B. Kleiner and S. Stadler.
Mar 8- Alex Margolis (Vanderbilt University)
A cornucopia of simple lattices in products of trees
The class of groups acting properly and cocompactly on the direct product of two regular locally finite trees is much richer than one might initially expect, as demonstrated by work of Wise and Burger-Mozes. Most notably, Burger-Mozes exhibited the first known examples of finitely presented infinite simple torsion-free groups within this class of groups.
In this talk, I will discuss a random model for investigating these groups. With high probability, the groups in this random model are just-infinite. I will also discuss a counting result on the number of virtually simple groups in this class, giving an abundance of virtually simple finitely presented groups. This is joint work with Nir Lazarovich and Ivan Levcovitz.
Mar 22-24 - Special departmental event: Rado Lecture (Richard Schoen)
Mar 24- Antonio Viruel (Malaga)
Inflexible Algebras
An oriented closed connected d-manifold is inflexible if it does not admit self-maps of unbounded degree. In addition, if for every oriented closed connected d-manifold M' the set of degrees of maps from M' to M is finite, then M is said to be strongly inflexible. The first examples of simply connected inflexible manifolds have been constructed by Arkowitz and Lupton using Rational Homotopy Theory. However, it is not known whether simply connected strongly manifolds exist, problem that is related to Gromov's question on functorial semi-norms on homology. In this talk, using Sullivan models, we present a method that proves the failure of strongly inflexibility for all but one of the existing inflexible manifolds. This is a joint work with Cristina Costoya and Vicente Muñoz.
Mar 29- Eduard Einstein (University of Pittsburgh)
Relative Cubulation of Small Cancellation Quotients
Daniel Groves and I introduced relatively geometric actions, a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex. Building on results of Martin and Steenbock for properly and cocompactly cubulated groups, Thomas Ng and I proved that C’(1/6)--small cancellation free products of relatively cubulable groups are relatively cubulable. The flexibility of relatively geometric actions allowed us to prove that C’(1/6)--small cancellation free products of residually finite groups are residually finite – without any need to assume that the factors are cubulable. In this talk, I will discuss techniques used to produce relatively geometric cubulations, applications to small cancellation quotients and potential future applications to random groups and small cancellation quotients of relatively hyperbolic groups.
Mar 31- Michael Falk (Northern Arizona University)
Models for braid-like groups and their pure subgroups
We describe a construction that produces a model for the braid group, starting from the graph of the permutahedron, and show how it can be generalized to graphic braid groups. The latter arise from consideration of fundamental groups of graphic hyperplane arrangements, which are ``pure graphic braid groups.’’ We describe some partial structure theorems for those groups, ending with an open problem. This talk is based on joint work with various people including Dan Cohen, Emanuele Delucchi, Dana Ernst, and Sonja Riedel.
April 4-6 - Special departmental event: Zassenhaus Lecture (Mladen Bestvina)
April 12- Giovanni Paolini (Caltech)
The K(π,1) conjecture
Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A long-standing open problem, called the K(π,1) conjecture, states that these configuration spaces are classifying spaces for the corresponding Artin groups. In the case of finite Coxeter groups, this was proved by Deligne in 1972. In the first part of this talk I will introduce Coxeter groups, Artin groups, and the K(π,1) conjecture. Then I will outline a recent proof of the K(π,1) conjecture in the affine case (a joint work with Mario Salvetti) and next directions.
April 14- Dave Penneys (OSU)
Unifying definitions of analytic approximation and rigidity properties for discrete groups, quantum groups, and unitary tensor categories
Unitary tensor categories are mathematical objects which simultaneously generalize discrete groups and their categories of finite dimensional Hilbert space representations. They arise naturally in the context of quantum mathematical objects like von Neumann algebras and topologically ordered phases of matter. In this talk, we will discuss analytic approximation and rigidity properties for unitary tensor categories, like amenability, the Haagerup property, and Kazhdan's property (T), and how our definitions give a simultaneous definition for discrete groups and quantum groups as well. We will assume no familiarity with unitary tensor categories.
April 19- Karen Butt (University of Michigan)
Quantitative marked length spectrum rigidity
The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. Conjecturally, the marked length spectrum determines the metric up to isometry (Burns-Katok). This is known to be true in some special cases, namely in dimension 2 (Otal, Croke), in dimension at least 3 if one of the metrics is locally symmetric (Hamenstadt, Besson-Courtois-Gallot), and in any dimension if the metrics are assumed to be sufficiently close in a suitable C^k topology (Guillarmou-Knieper-Lefeuvre). Even in these cases, there is more to be understood about to what extent the marked length spectrum determines the metric. Namely, if two manifolds have marked length spectra which are not equal but are close, is there some sense in which the metrics are close to being isometric? In this talk, we will provide some (quantitative) answers to this question, refining the known rigidity results for surfaces and for locally symmetric spaces of dimension at least 3.