Topology and Geometric Group Theory Seminars at the Ohio State University

Aug 27 -

Sep 1 - 

Sep 3 - Francesco Lin (Columbia)

Adjunction inequalities and the Davis hyperbolic four-manifold

LeBrun conjectured that the Seiberg-Witten invariants of closed hyperbolic four-manifolds vanish. After discussing context and consequences, I will provide an overview of what is known, and then focus on the proof (joint work with Bruno Martelli) that the conjecture holds for the Davis manifold.

Sep 8 - 

Sep 10 - Yu-Chan Chang (OSU)

Isomorphism Problems for Bestvina--Brady Groups

Bestvina--Brady groups are subgroups of right-angled Artin groups, and both families of groups are defined on finite simple graphs. This talk will address two isomorphism problems concerning Bestvina--Brady groups. Specifically, I will discuss how that tree spanners and matroids come into play when answering these questions. This is joint work with Lorenzo Ruffoni.

Sep 15 - 

Sep 17 - Maya Verma (OSU)

Torsion-free lattices in Baumslag-Solitar complexes 

For the Cayley 2-complex X_{m,n}​ of a Baumslag-Solitar group BS(m,n), one can consider the group of combinatorial automorphisms, denoted Aut(X_{m,n}​). In this talk, we explore torsion-free uniform lattices in the locally compact group Aut(X_{m,n}​). We classify the pairs of nonzero integers (m,n) for which Aut(X_{m,n}​) contains torsion-free incommensurable lattices.

Sep 22 - 

Sep 24 - Yukun Du (University of Georgia)

Two-Generator Discrete Subgroups of Tree Automorphism Groups

I will discuss the classification of discrete two-generator groups acting on locally finite trees. Under a small-geometry condition, the possible groups and quotient graphs are strongly constrained and can be classified explicitly. I will then explain how the picture becomes much richer beyond this regime, including higher-index amalgams and families with arbitrarily complicated quotient geometry.

Sep 29 - 

Oct 1 -

Oct 6 -

Oct 8 - Zhihao Mu (CUNY)

Connectivity of boundaries of right-angled Coxeter groups 

A central theme in geometric group theory is to study finitely generated groups through their large-scale geometry. Right-angled Coxeter groups (RACGs) are defined by finite graphs, and much of their large-scale geometry is visible in the combinatorics of these defining graphs. To each RACG, we can canonically associate a hyperbolic space, obtained by coning-off flats in its Davis complex. In this talk, I will characterize, in terms of combinatorics of the defining graph, when the Gromov boundary of this hyperbolic space is connected. Moreover, whenever the boundary is connected, it is also linearly connected. As an application, among one-ended RACGs, the existence of a splitting over a subgroup contained in a direct product of two infinite subgroups is preserved by quasi-isometries. 

Oct 13 - Chloé Postel-Vinay (U Chicago)

k-shuffle polynomials

One of the ways that braid groups manifest themselves is as fundamental groups of spaces of monic, complex, degree n polynomials with distinct roots. Another way to phrase this is: polynomials for which 0 is a regular value. In this talk, instead on focusing on just 0 being regular, we study spaces of polynomial for which the k roots of unity are regular values. Their fundamental groups are called k-shuffle braid groups (for which we will give a presentation), and these spaces also turn out to be classifying spaces. Finally, if time permits, we will see how k-shuffle braid groups inject into larger braid groups through polynomial homotopies.

Oct 15 - Autumn Break

Oct 20 - 

Oct 22 - Matt Zaremsky (SUNY Albany)

Finitely presented simple groups, and the state of the art of the Boone-Higman conjecture 

In the 1970's, Boone and Higman conjectured that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this talk I will discuss recent progress on this conjecture, and highlight the important role played by a certain family of generalizations of Thompson's group V. In particular, in recent joint work with Francesco Fournier-Facio and Xiaolei Wu, we prove that whenever a group satisfies the Boone-Higman conjecture, it also satisfies a related embedding conjecture entirely in the world of Thompson-like groups. This talk will also touch on joint work with Jim Belk and James Hyde.

Oct 27 - Meenakshy Jyothis (U of Oklahoma)

Stars at infinity in the Thurston boundary 

For a complete metric space with a chosen compactification, the star of a boundary point is a subset of the boundary determined by the metric and the compactification. Karlsson introduced stars to study the dynamics of isometries. For instance, under suitable hypotheses, iterates of an isometry carry points outside the repelling based star toward the attracting based star, giving a generalized north–south behavior.

In this talk, we study stars in the Thurston compactification of Teichmüller space with respect to Thurston’s metric and the Teichmüller metric. We show that the stars for these metrics differ, disproving Karlsson’s conjecture for the Teichmüller metric. For surfaces of complexity at least two, we show that the hyperbolic boundary points for both metrics are precisely the projective classes of uniquely ergodic measured laminations. This is joint work with Jing Tao.

Oct 29 - 

Nov 3 - Manisha Garg (UIUC)

Geometry of groups through their boundaries 

Geometric group theory studies groups through their large-scale geometry. Boundaries at infinity capture the asymptotic behavior of geodesics and reveals important geometric and algebraic properties of groups. In this talk, I will ask which features of a hyperbolic group’s large-scale geometry remain detectable when we change the metric on its boundary. I will then move beyond hyperbolicity to sublinear Morse boundaries and introduce a sublinear metric based on the fellow-traveling behavior of geodesics (ongoing joint work with Sagnik Jana and Yulan Qing). Returning to the hyperbolic setting, I will discuss its geometric properties, its relationship with polynomial Floyd metrics, and questions about recovering the large-scale geometry of groups from their boundaries. 

Nov 5 - Emily Stark (Wesleyan)

Conformal dimension for boundaries of certain Coxeter groups 

Conformal dimension is a powerful analytic invariant of a metric space that captures the fractal behavior amongst deformations of the metric space. This invariant has made a significant impact on geometric group theory and the study of spaces with negative curvature. We study a family of Coxeter groups that are built of Kleinian groups and are hyperbolic relative to virtually abelian subgroups. We give bounds on the conformal dimension of the Bowditch boundary of these groups. Our results imply there are infinitely many quasi-isometry classes within the family of Coxeter groups with defining graph a complete graph and edge labels greater than or equal to three. Consequently, there are infinitely many quasi-isometry classes among hyperbolic groups with Pontryagin sphere boundary. This is joint work with Elizabeth Field, Radhika Gupta, and Robbie Lyman. 

Nov 10 - 

Nov 12 - Christopher Cericola (Oregon State)

Developments in crossing change operations

In 2014 Ayaka Shimizu published a paper containing the region crossing change. This operation on knot diagrams proved useful in building relations between knots and their invariants. The arc crossing change is a new operation on knot diagrams, but it is heavily inspired by Shimizu's work. In this talk, I will describe some of the developments that have extended Shimizu's results to the arc crossing change, and the direction that people have taken those results. 

Nov 17 - Rado Lectures

Nov 19 - 

Nov 24 - Amy Tao (UW-Madison)

Nov 26 - Thanksgiving break

Dec 1 - Andrey Gogolyev (OSU)

Dec 3 - 

Dec 8 -