Topology and Geometric Group Theory Seminars at the Ohio State University

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)

Feb 6 - 

Feb 11 - Annette Karrer (OSU)

Feb 13 - 

Feb 18 - 

Feb 20 - 

Feb 25 - 

Feb 27 - 

Mar 4 - Jean Lafont (OSU)

Mar 6 - 

Mar 18 - 

Mar 20 - Katherine Goldman (McGill)

Mar 25 - 

Mar 27 - 

April 1 - 

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April 8 - 

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April 15 - 

April 17 - Michael Dougherty (Lafayette College)

April 22- Jonathan Zung (MIT)

April 29 -Bin Sun (Michigan State University)