Spring 2021
Meeting dates: Tuesdays (Topology) and Thursdays (Geometric Group Theory) from 1:00 to 2:00 pm ET
Seminar organizers - Jingyin Huang, Christoforos Neofytidis, Mark Pengitore, and Rachel Skipper
Spring 2021 Schedule
The seminars will be run virtually over Zoom through the Spring 2021 Semester
https://osu.zoom.us/j/93661626526?pwd=b2xiSEJSTm9BRVlRSitOZXVkMVMzZz09
Zoom ID: 936 6162 6526, Password: 273789
January 12 -
Christoforos Neofytidis (OSU)
Endomorphisms of mapping tori
One of the most fundamental results in 3-dimensional topology, proved in works of Gromov, Mostow, Wang and Waldhausen, is that any self-map of non-zero degree of a mapping torus of a closed hyperbolic surface is homotopic to a homeomorphism if and only if the monodromy is not periodic. Key properties for the proof were the existence of hyperbolic structures or of non-vanishing semi-norms (such as the simplicial volume). Using Algebra, we give a new, unified proof and generalise the above result in every dimension, by replacing the hyperbolic surface with a corresponding higher dimensional aspherical manifold. More generally, we will classify in terms of Hopf-type properties mapping tori of residually finite Poincaré Duality groups with non-zero Euler characteristic. It turns out that the rigidity behavior of these mapping tori with trivial center is similar to that of non-elementary torsion-free hyperbolic groups.
January 14 -
January 19 -
Mike Davis (OSU)
Bordifications of hyperplane arrangement complements and curve complexes of spherical Artin groups
The complement of an arrangement of hyperplanes in a complex vector space has a natural bordification to a manifold with corners formed by removing tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of the braid group. The faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex, the vertices of which are the irreducible ``parabolic subgroups'' of the fundamental group of the arrangement complement. When the arrangement is associated to a finite reflection group, we get the "curve complex" of the associated pure Artin group. In analogy with curve complexes for mapping class groups and with spherical buildings, our curve complex has the homotopy type of a wedge of spheres. This is joint work with Jingyin Huang.
January 21 -
Daniel Berlyne (City University of New York)
Graph products as hierarchically hyperbolic groups
Given a finite simplicial graph with a finitely generated group associated to each vertex, the graph product is defined by taking the free product of the vertex groups and adding commutation relations between elements belonging to vertex groups that are connected by a edge in the graph. Common examples of graph products include right-angled Artin groups (where all vertex groups are Z) and right-angled Coxeter groups (where all vertex groups are Z/2Z). Behrstock, Hagen, and Sisto showed that right-angled Artin groups exhibit a notion of non-positive curvature called hierarchical hyperbolicity, with deep geometric consequences such as a Masur-Minsky style distance formula, finite asymptotic dimension, and acylindrical hyperbolicity. By developing analogues of the cubical techniques employed by Behrstock-Hagen-Sisto, we are able to generalise their result, showing that any graph product with hierarchically hyperbolic vertex groups is itself a hierarchically hyperbolic group. In doing so, we answer two questions of Behrstock-Hagen-Sisto and two questions of Genevois. This is joint work with Jacob Russell.
January 26 -
Francis Wagner (Vanderbilt)
Torsion Subgroups of Groups with Quadratic Dehn Function
The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that this property does not carry over to any class of groups of larger Dehn function. In particular, for every m>1 and n sufficiently large (and either odd or divisible by 2^9), there exists a quasi-isometric embedding of the infinite free Burnside group B(m,n) into a finitely presented group with quadratic Dehn function.
January 28 -
Panagiotis Konstantis (Marburg)
GKM manifolds - Interactions between combinatorics and topology
A GKM manifold is a smooth manifold endowed with a certain type of Torus action. To every GKM manifolds one assigns a combinatorial object, the GKM graph, which encodes important properties of the torus action. We discuss how far this object determines the topology and the smooth structure of a GKM manifold. This is joint work with Oliver Goertsches and Leopold Zoller.
February 2 -
David Hume (Bristol)
Coarse Geometry of Groups and Spaces
Given two metric spaces X and Y it is natural to ask how faithfully, from the point of view of the metric, one can embed X into Y.
One way of making this precise is asking whether there exists a coarse embedding of X into Y.
Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. In particular I will show that the Baumslag-Solitar group BS(m,n) coarsely embeds into some hyperbolic group if and only if |m|=|n|=1.
February 4 -
Teddy Einstein (UIC)
Relatively Geometric Actions on CAT(0) Cube Complexes
The study of hyperbolic and relatively hyperbolic groups acting on CAT(0) cube complexes has produced exciting recent results in geometric group theory. I will talk about a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex called a relatively geometric action.
In joint work with Daniel Groves, we develop analogues of tools used to construct and study geometric actions of hyperbolic and relatively hyperbolic groups on CAT(0) cube complexes, including a relatively geometric version of Agol's Theorem.
I will also discuss some of the structural theorems we hope to prove and a potential application to the Relative Cannon Conjecture.
February 9 -Note special time 6:00 pm
Yash Lodha (KIAS)
Spaces of enumerated orderable groups
An enumerated group is a group structure on the natural numbers.
Given one among various notions of orderability of countable groups, we endow the class of orderable enumerated groups with a Polish topology.
In this setting, we establish a plethora of genericity results using elementary tools from Baire category theory and the Grigorchuk space of marked groups.
In this talk I will describe these spaces and some of their striking features.
This is ongoing joint work with Srivatsav Kunnawalkam Elayavalli and Issac Goldbring.
February 11 -
Yulan Qing (Fudan University)
Sublinearly Morse Boundary of Groups
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 $\kappa$-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 mainly based on several joint projects with Ilya Gekhtman, Kasra Rafi and Giulio Tiozzo.
February 16 -
Roman Sauer (KIT)
Action on Cantor spaces and macroscopic scalar curvature
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of l2-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of $1$-balls in the universal cover. Group actions on Cantor spaces surprisingly play an important role in the proof. The talk is based on joint work with Sabine Braun.
February 18 -
February 25 -
Stefan Witzel (Giessen)
Uncountably many simple groups up to quasi-isometry
Abstract: The purpose of geometric group theory is to investigate groups up to quasi-isometry, a coarse geometric notion. Many classes of groups contain uncountably many finitely generated groups up to isomorphism. From a geometric perspective one is led to ask (for each class) whether this remains true up to quasi-isometry. I will talk about joint work with Ashot Minasyan and Denis Osin where we use the Baire category theorem to answer such questions. Specifically I will show that thereare uncountably many finitely generated simple groups up to quasi-isometry.
March 2 -
March 4-
Rostislav Grigorchuk (Colloquium at 4:15 pm ET)
Groups, growth and spectra
I will give an introduction to some asymptotic invariants of finitely generated groups and their applications to geometry, random walks and topology. This talk will be accessible to graduate students and non-experts.
March 9 - Note special time 6:30 pm
Makoto Sakuma (Osaka City University and Hiroshima University)
Homotopy motions of surfaces in 3-manifolds
We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behaviour of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting. This is a joint work with Yuya Koda (arXiv:2011.05766).
March 11 -
March 16 -
Aaron Calderon (Yale University)
Measure laminations and unipotent flows on moduli space
There is a deep yet mysterious connection between the hyperbolic and singular flat geometry of Riemann surfaces. Using Thurston and Bonahon’s “shear coordinates” for maximal laminations, Mirzakhani related the earthquake and horocycle flows on moduli space, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. In this talk, I will describe joint work with James Farre in which we construct new coordinates for Teichmüller space adapted to any measured lamination which generalize both Fenchel–Nielsen and shear coordinates. These coordinates simultaneously parametrize both flat and hyperbolic structures, and consequently allow us to extend Mirzakhani’s conjugacy and gain insight into the ergodic theory of the earthquake flow. If time permits, I will also mention some applications of this result to the equidistribution of random hyperbolic surfaces in moduli space.
March 18 -
Jason Behrstock (City University of New York)
Hierarchically hyperbolic groups: an introduction
Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, most cubulated groups, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view, both describing new tools to use to study these groups and applications of those results. This talk will include joint work with Mark Hagen and Alessandro Sisto.
March 23 -
Stephan Mescher (Leipzig)
Topological complexity of manifolds dominated by products
Topological complexity (TC) was introduced by M. Farber as an integer-valued homotopy invariant motivated by the motion planning problem from robotics. After a short introduction to TC and an overview of basic results, I will discuss how the cohomology rings of a space can be employed to derive lower bounds on TC. I will present a computation of TC for certain symplectic manifolds. For general closed manifolds, I will further outline a connection between TC and the domination of manifolds by products of the form M x S^1. If time permits, I will outline some approaches to the computation of TC of aspherical spaces as well.
March 25 -
Andy Putman (Notre Dame)
The mapping class group of connect sums of S^2 \times S^1
Strengthening a classical theorem of Laudenbach, I will explain how to prove that the mapping class group of the 3-manifold obtained as the connect sum of n copies of S^2 \times S^1 splits as a semidirect product of Out(F_n) by a finite abelian group of "sphere twists".
This is joint work with Tara Brendle and Nate Broaddus.
March 30 -
April 6 -
Hung Cong Tran (University of Oklahoma)
Superexponential Dehn functions inside CAT(0) groups
We construct 4--dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^m)$ for integers $n, m \geq 1$ and 6--dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^\alpha)$ for integers $n \geq 1$ and $\alpha$ dense in $[1,\infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups. This is a joint project with Noel Brady.
April 8 -
April 13 -
Tian-Jun Li (U Minnesota)
Symmetries of symplectic rational surfaces
We study symplectic rational surfaces equipped with a finite symplectomorphism group G. They are the symplectic analog of (complex) rational G-surfaces studied in algebraic geometry, which are rational surfaces equipped with a holomorphic G-action. These rational G-surfaces played a central role in the classification of finite subgroups of the plane Cremona group, a problem dating back to the early 1880s. Our work shows that a large part of the story regarding the classification of rational G-surfaces can be recovered by techniques from 4-manifold theory and symplectic topology. Furthermore, we also add some new symplectic geometry aspect to the study of rational $G$-surfaces. This is a joint work with Weimin Chen and Weiwei Wu.
April 15 -
Tim Riley (Cornell)
Subgroup distortion in hyperbolic groups
I will give a tour of the world of subgroups of hyperbolic groups through the lens of distortion---that is, how different a subgroup's own word metric is from that of the group in which it sits.
April 20 -
Jim Fowler (OSU)
Free group actions on $S^n \times S^n$
In 1925, Hopf stated the spherical space form problem: which groups act freely on a sphere? This was solved in the late 70s, but an extension of the problem remains open: which groups act freely on $S^n \times S^n$? It is known that a group acting freely on $S^n \times S^n$ cannot contain $A_4$ or $(\mathbb{Z}/p)^3$ for $p > 3$. We'll describe other obstructions, and for the special case of $(\mathbb{Z}/p)^2$ acting freely on $(S^n \times S^n)$, we will discuss the classification of the quotients.
April 22 - Note special time 11:30 am
Nicolas Matte-Bon
Confined subgroups and highly transitive actions
A subgroup of a group is confined if its conjugacy class in the Chabauty space does not accumulate on the trivial subgroup; this is tightly related to the notion of uniformly recurrent subgroup (URS) of Glasner and Weiss. In this talk I will explain a connection between confined subgroups and highly transitive actions, which are actions that are transitive on the set of distinct k-tuples for every k>0. We will see how this can be used as a tool to understand highly transitive actions of certain groups, or to rule out their existence.
This is joint work with Adrien Le Boudec.