Autumn 2022
Meeting dates: Tuesdays (Geometric Group Theory) 1:50-2:50 pm at Journalism Building 0291 and
Thursdays (Topology) from 1:50 to 2:50 pm at Caldwell Laboratory 0135
Organizers - Arman Darbinyan, Jingyin Huang, Heejoung Kim, Christoforos Neofytidis, and Francis Wagner
Autumn 2022 Schedule
The seminars will be run partially in person and partially virtually over Zoom for the Autumn 2022 Semester
https://osu.zoom.us/j/94449451323?pwd=RXZ5am9hYzVlaHVEaGNNRnpMbkRsQT09
Meeting ID: 944 4945 1323 Password: 779408
September 20-Jean-François Lafont (OSU)
Barycenter method in bounded cohomology
The barycenter map technique was pioneered by Besson-Courtois-Gallot in the early 1990s, and provides a powerful tool for establishing rigidity theorems. I'll explain how the technique can be used to analyze the comparison map from bounded cohomology to ordinary cohomology, and will survey some of the results obtained via this technique.
October 11- Sam Nariman (Purdue University)
(Un)boundedness of characteristic classes of manifold bundles.
The plane bundles over surfaces are classified by their Euler class and Milnor showed that if they admit flat structures, the Euler class should be a bounded class. This result has been generalized in many directions and in particular for a connected finite-dimensional Lie group G, it has been extensively studied the conditions under which the characteristic classes of flat principal G-bundles are bounded classes viewed in the cohomology of the classifying space of the group G with the discrete topology. In this talk, I will talk about a joint work with N.Monod, in which we started studying the same question for diffeomorphism groups in different regularities that are infinite-dimensional topological groups. In particular, we computed the bounded cohomology of Diff_0(S^1) and Diff_0(D^2) entirely and showed that they are isomorphic to the polynomial ring generated by the Euler class. We also answered Ghys’ question about generalizing Milnor-Wood inequality to flat S^3-bundles by showing that the Euler class is unbounded in H^4(Diff_0(S^3);R).
October 18- Wenhao Wang (Steklov Mathematical Institute)
Orders on Free Metabelian Groups
A bi-order on a group $G$ is a total, bi-multiplication invariant order. Such an order is regular if the positive cone associated to the order can be recognised by a regular language. A subset $S$ in an orderable group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this talk, I will discuss the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order. As an application, I prove that non-abelian free metabelian groups of finite rank do not admit a regular bi-order while they are computably bi-orderable.
October 20-Rose Morris-Wright (Middlebury College)
Rewrite systems in 3-free Artin groups
(Joint work with Maria Cumplido and Ruben Blasco)
Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. In particular, it is unknown whether the word problem is solvable for all Artin groups. I will discuss a new algorithm for solving the word problem in 3-free Artin groups. This is based on work by Holt and Rees for large type and sufficiently large type groups (2012 and 2013). Our work significantly broadens the class of Artin groups with solvable word problem. This algorithm gives an explicit way to reduce a word to a geodesic form without ever increasing the length of the word.
October 27-Srivatsav Kunnawalkam Elayavalli (UCLA)
Sofic approximations of amenable groups
I will present a recent Theorem of mine and B. Hayes that every non amenable group which is a limit of amenable groups in the space of marked groups has two sofic embeddings that are not equivalent by any automorphism of the universal sofic group. This generalizes a theorem of Elek-Szabo from 2010 and addresses a question of Paunescu.
November 10-Yandi Wu (University of Wisconsin)
A topologically rigid set of quotients of the Davis complex
A class of topological spaces is topologically rigid if for any pair of spaces in the class, an isomorphism on the level of fundamental groups induces a homeomorphism. One famous example of a topologically rigid class is the set of simply connected 3-manifolds (Poincare’s conjecture). Lafont also proved a series of results about topological rigidity of hyperbolic piecewise manifolds, which have played a big role in my work. In this talk, we examine a set of K(G,1) spaces of certain right-angled Coxeter groups (RACGs), which is not a topologically rigid class. However, if we impose certain restrictions on the defining graphs of the RACGs, we can obtain an infinite topologically rigid subclass.
November 15-Justin Katz (Purdue University)
Spectral rigidity of some hyperbolic 2 and 3 manifolds
In this talk, I will discuss a result contained in my graduate thesis: that certain arithmetic hyperbolic Riemann surfaces and 3 manifolds are uniquely determined by their Laplace eigenvalue spectrum. If time permits, I will also describe a program for proving rigidity results of broader scope. This work is joint with D.B. McReynolds."
November 22-Peng Hui How (University of Chicago)
Simplicial Volume of Closed Locally Homogeneous Riemannian Manifolds
Let M be a closed smooth manifold. Two natural questions on include: 1. Is Minvol(M)=0? (we will define Minvol) 2. Does M admit a self-map of degree >1 ? In the paper "Volume and Bounded Cohomology", Gromov defined the simplicial volume ||M||, which is a non-negative topological invariant. ||M||>0 is an obstruction to 1. and 2. A remarkable paper by LaFont and Schmit showed that ||M||>0 when is a closed locally symmetric space of non-compact type. We showed that the converse holds in the scope of closed locally homogeneous Riemannian manifolds (CLHRM). Meanwhile, even for CLHRM, as far as we understand, 1. and 2. are still open in many cases, and we are still working on it. In the talk, we will discuss 1. and 2. with many examples, and the idea of the proof of our result.
November 22-Thomas Haettel (University of Montpellier)
Group actions on nonpositively curved simplicial complexes
I will focus on metric spaces which admit a convex unique bicombing (CUB) spaces, which generalize notably CAT(0) spaces. I will present a link condition for simplicial complexes stating essentially that the links of vertices are lattices. I will present some applications of this criterion, which include notably Euclidean buildings, braid groups, Garside groups, Deligne complexes of some Artin groups, some curve complexes.
Special event: Rado Lecture series from Nov. 30 - Dec. 2
December 1- Nathan Broaddus (OSU)
The Steinberg Module of the braid group
The braid group on n strands is a Bieri-Eckmann duality group due to the fact that it is torsion free and has a finite index subgroup (the pure braid group) which is an iterated extension of free groups by free groups. I will talk about current joint work with Lindsey-Kay Lauderdale (Southern Illinois University), Emille Lawrence (University of San Francisco), Anisah Nu'Man (Spelman College) and Robin Wilson (Loyola Marymount University) in which we seek to improve upon the known presentations of the dualizing module of the braid group as a braid group module.
December 6-Arman Darbinyan(OSU)
A quasi-isometrically diverse class of left-orderable solvable and residually finite groups.
In my talk I will discuss a new uncountable class of pairwise non-quasi-isometric groups that are two-generated, solvable of derived length 3, left-orderable, and residually finite. In particular, this provides the first example of quasi-isometrically diverse class of left-orderable groups, and, in addition, recovers or strengthens several known results in this direction that will also be reviewed. To achieve this result, we introduce a new quasi-isometric invariant for groups that we call arc-completion spectra. The talk will be based on an ongoing project of mine.
Special event: Minicourse on Out(F_n)
December 6 - Camille Horbez (Université Paris-Saclay)
Introduction to Out(Fn) and Outer space I
The lectures will be an introduction to the group Out(Fn) of outer automorphisms of a finitely generated free group, and to Outer space. The latter was introduced by Culler and Vogtmann in 1986, in analogy to the Teichmüller space of a surface; it is a contractible space on which Out(Fn) acts with finite stabilizers. The lectures will focus on the topology (e.g. contractibility) and geometry (metric aspects and negative curvature features) of Outer space, and I will explain how these features can be used to deduce some algebraic properties of Out(Fn).
December 8 - Camille Horbez (Université Paris-Saclay)
Introduction to Out(Fn) and Outer space II
December 12 - Camille Horbez (Université Paris-Saclay)
Introduction to Out(Fn) and Outer space III