Jan 7 - Yanlong Hao (UMichigan)

Symmetry vs Arithmeticity: Coarse geometry perspective

Margulis's theorem asserts that a lattice in a semisimple, center-free Lie group without a compact factor is arithmetic if its commensurator is dense. This result has been further developed into a geometric framework. In this talk, we aim to generalize these works to the setting of coarse geometry.

Jan 9 - 

Jan 14 - 

Jan 16 - Alex Margolis (OSU)

 Coarse homological invariants of metric spaces

A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or infinity. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and Hk(G,FG)=0 for k<n, then dim Hn(G,FG)=0,1 or ∞, significantly extending work of Farrell from 1975. Moreover, in the case dim Hn(G,FG)=1, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces.

In this talk, we talk about the tools needed to prove this result. We will introduce several coarse topological invariants of metric spaces, inspired by group cohomology. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group equipped with a proper left-invariant metric, then the coarse cohomological dimension of G coincides with its cohomological dimension whenever the latter is finite. Extending a result of Sauer, we show that coarse cohomological dimension is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one.   

Jan 21 - Hyeran Cho (OSU)

Hyperbolicity of Random Branched Coverings

For a finitely presented group $\Gamma$ with a finite presentation, let $X$ be the presentation $2$-complex. We introduce $n$-fold random branched coverings of $X$ branched over the centers of its $2$-cells. Especially for the finitely generated one-relator group case with a single primitive relator, we prove that fundamental groups of random branched coverings are asymptotically almost surely Gromov hyperbolic. In other words, for a random branched covering $X(\sigma)\rightarrow X$, the probability that $\pi_1(X(\sigma))$ is Gromov hyperbolic goes to $1$ in the limit $n\rightarrow\infty$.

Jan 23 - 

Jan 28 - Rima  Chatterjee (OSU)

Classification of knots vs. links in the contact world

A contact 3- manifold is a smooth manifold equipped with a special geometric structure. When we think of a knot in it, it also comes with a special geometric structure.  A knot is called Legendrian if it is everywhere tangent to the contact planes.  In contact world, classification of knots are lot finer than that of the smooth world. In this talk, I will give a gentle introduction to contact manifolds and knots in them and discuss some of the interesting classiifcation results. Time permitting, I'll mention how the classification gets extremely hard and interesting, when one starts considering links in contact manifolds. I will not assume any prior knowledge of contact topology. Everyone is welcome!

Jan 30 -

Feb 4 - George Domat (U Michigan)

Classification of Stable Surfaces with respect to Automatic Continuity

Topological groups often exhibit lots of interplay between their algebraic and topological structures. A stark example of this is the automatic continuity property: A topological group has the automatic continuity property if every (algebraic) homomorphism to any other separable group is continuous. We provide a complete classification of when the homeomorphism group of a stable surface has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. This is joint work with Mladen Bestvina and Kasra Rafi.

Feb 6 - 

Feb 11 - 

Feb 13 - Colloquium talk by Alex Wright (U Michigan)

Feb 18 - 

Feb 21 - Barry Minemyer (Commonwealth University - Bloomsburg)

 Complex Hyperbolic Gromov-Thurston Metrics

 In 1987 Gromov and Thurston developed the first Riemannian manifolds that are not homotopy equivalent to a hyperbolic manifold, but admit a Riemannian metric that is ϵ-pinched for any given ϵ>0. The manifolds that they construct are branched covers of hyperbolic manifolds, and to construct the metric they perform a sort of "geometric surgery" about the ramification locus. In 2022 Stover and Toledo proved the existence of similar branched cover manifolds built out of complex hyperbolic manifolds, and via a result of Zheng these manifolds admit a negatively curved Kahler metric. In this talk we will discuss how to construct a (not Kahler) Riemanain metric on these Stover-Toledo manifolds which is ϵ-close to being negatively 1/4-pinched for any prescribed ϵ>0.  

Feb 27 - 

Mar 4 - Jean Lafont (OSU)

Kent-Leininger 4-manifolds with vanishing signature

The Kent-Leininger construction provides atoroidal 4-manifolds that are surface bundles over surfaces. I'll explain why some of these examples have vanishing signature. This is joint work with Nick Miller and Lorenzo Ruffoni.

Mar 6 - 

Mar 18 - 

Mar 20 - Katherine Goldman (McGill)

Residual properties of 2-dimensional Artin groups

It is a longstanding open question to determine which Artin groups are residually finite. Past results have followed from linearity (e.g., for spherical-type) or product decompositions in rank 3. We present a new approach to this problem using intermediate quotients to so-called Shephard groups. These Shephard groups possess their own interesting (and sometimes counterintuitive) geometry which we can leverage to give new information about their corresponding Artin groups in some cases. As a highlight of this connection, we show that an Artin group which is simultaneously 2-dimensional, hyperbolic-type, and FC-type is residually finite. One of the key features of the proof we will discuss is the fact that hyperbolic-type 2-dimensional Shephard groups are relatively hyperbolic, which is almost never true of Artin groups.

Mar 25 - 

Mar 27 - Anette Karrer (OSU)

Connected Components in Morse boundaries of right-angled Coxeter groups

Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isometric to right-angled Coxeter groups.  This talk is based on two projects.  One is joint work with Bobby Miraftab and Stefanie Zbinden. The other one is joint work in progress with Matthew Cordes and Kim Ruane.

April 1 - 

April 3 - 

April 8 - Mike Davis (OSU)

Exotic aspherical 4-manifolds

In joint work with Hayden, Huang, Ruberman, and Sunukjian we show that there are  closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic.

April 10 - Pallavi Dani (Louisiana State University)

Subgroup distortion in hyperbolic groups

The distortion function of a subgroup measures the extent to which the intrinsic word metric of the subgroup differs from the metric induced by the ambient group. Ol’shanskii showed that there are almost no restrictions on which functions arise as distortion functions of subgroups of finitely presented groups. This prompts one to ask what happens if one forces the ambient group to be particularly nice, say, for example, to be hyperbolic. I will survey which functions are known to be distortion functions of subgroups of hyperbolic groups and describe joint work with Tim Riley in which we construct new examples of such functions.

April 15 -  Yuping Ruan (Northwestern)

Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has nontrivial Euclidean de Rham factors, or it has positive simplicial volume. This is based on a joint work with Chris Connell and Shi Wang, arXiv:2410.19981.

April 17 - Michael Dougherty (Lafayette College)

Cell Structures for Braid Groups from Complex Polynomials

In this talk, I will describe a new geometric and combinatorial structure for the space of complex polynomials with a fixed number of roots. In particular, I will define a metric on the space of monic polynomials with d distinct centered roots, and I will introduce a finite cell structure for the metric completion. Each cell in this complex is a product of two Euclidean simplices, and the combinatorial structure comes from the dual presentation for the d-strand braid group. In particular, this provides a concrete connection between two classifying spaces for the braid group. This is joint work with Jon McCammond.

April 22- Jonathan Zung (MIT)

                                                                                 Expansion and torsion homology of 3-manifolds

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated: they must have lots of torsion homology.

April 29 -Bin Sun (Michigan State University)

Cohomological dimensions of finitely generated groups and their subgroups

I will present recent joint work with Francesco Fournier-Facio. Given a group G, consider the set S(G) of cohomological dimensions of its subgroups. We prove that, with obvious exceptions, every subset of N∪ {∞} can be realized as S(G) for some group G. 

As an application, we answer a question of Talelli in the negative: there exists a torsion-free group G of infinite cohomological dimension such that all proper subgroups of G have uniformly bounded finite cohomological dimensions. Our construction also yields the first examples of torsion-free Smith groups–groups whose actions on CW-complexes always have global fixed points. Moreover, by analyzing the L2-Betti numbers of our examples, we obtain the first uncountable family of mutually non-measure equivalent, finitely generated, torsion-free groups.