Topology and Geometric Group Theory Seminars at the Ohio State University

Jan 11 - Jean Lafont (OSU)

Totally geodesic submanifolds in hyperbolic hybrids

I will review  Gromov and Piatetski-Shapiro's hybridization construction of non-arithmetic hyperbolic manifolds. I will explain why these hyperbolic manifolds contain only finitely many codimension one totally geodesic submanifolds. This was joint work with D. Fisher, N. Miller, and M. Stover. 

Jan 16 - Lei Chen (University of Maryland) 

Mapping class groups of circle bundles over a surface

In this talk, we study the algebraic structure of mapping class group Mod(M) of 3-manifolds M that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku.

Jan 18 - Mitul Islam (Max-Planck Institute of Mathematics at Leipzig)

Morse-ness in convex projective geometry

The (Hilbert metric) geometry of properly convex domains generalizes real hyperbolic geometry. This generalization is far from the Riemannian notion of non-positive curvature, but they have some intriguing similarities. I will explore this connection from a coarse geometry viewpoint.

I will focus on Morse geodesics ("negatively curved directions", in a coarse sense) in properly convex domains. I will show that Morse-ness can be characterized entirely using linear algebraic data (i.e. singular values of matrices that track the geodesic). Further, I will discuss how this coarse geometric notion of Morse is related to the symmetric space geometry as well as the smoothness of boundary points. This is joint work with Theodore Weisman.

Jan 25 - Nima Hoda (Connell)

Tree of graph boundaries of hyperbolic groups

Regular trees of graphs are inverse limits of particularly simple inverse systems of finite graphs.  They form a 1-dimensional subclass of the Markov compacta: a class of finitely describable inverse limits of simplicial complexes, which includes all boundaries of hyperbolic groups.  I will discuss upcoming joint work with Jacek Swiatkowski in which we use Bowditch's canonical JSJ decomposition to characterize the 1-ended hyperbolic groups whose boundaries are (regular) trees of graphs.

Feb 13 - Indira Chatterji (Université Côte d'Azur)

Graphs of groups, distortion and the Rapid Decay property.

I will define and discuss fundamental groups of a graph of groups, how to control subgroups distortion in those and how it affects the rapid decay property, that I will also define and discuss.

Feb 15 - Mahan Mj (Colloquium)

Feb 29 - Fan Ye (Harvard)

Towards isomorphisms among Floer homologies

 Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds.  In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will first introduce the history of the constructions in Floer theory and then introduce an approach to proving the isomorphisms. The idea is based on combinatorial version of Floer homology, which leads to an axiomatic construction of Floer homology. This work is joint with Baldwin, Li, and Sivek.

Mar 5 - Aleksandar Milivojevic (Waterloo)

Poincaré dualization and Massey products

Rational homotopy theory allows us, in many situations, to move between the homotopy theory of spaces modulo torsion and that of commutative differential graded algebras. I will describe a construction that completes a commutative differential graded algebra, over a field of characteristic zero, to one satisfying Poincaré duality on its cohomology, and discuss when the construction is functorial. We can then apply this to the setting of non-zero degree maps between Poincaré duality spaces. By a result of Taylor, non-trivial triple Massey products remain non-trivial under pullback by a non-zero degree map. We will see that quadruple (and higher) Massey products are not so well behaved. This is joint work with Jonas Stelzig and Leopold Zoller. Time allowing, we will relate this story to the behavior of Massey products under field extension. 

Mar 19 -  Damian Osajda (University of Copenhagen)

Coxeter groups are biautomatic

We prove that all Coxeter groups are biautomatic. A biautomatic structure is obtained by constructing a new language - the voracious language. Based on a joint work with Piotr Przytycki (McGill).

Mar 26 - Ying Hu (University of Nebraska Omaha)

Title: Stir fry Homeo_+(S^1) representations from pseudo-Anosov flows 

Abstract: A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a $3$-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on $3$-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of $3$-manifold groups.  

In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n.  This is joint work with Steve Boyer and Cameron Gordon.

Mar 28 - 

April 2 - 

April 4 - Nick Salter (Notre Dame)

The equicritical stratification and stratified braid groups

Thinking about the configuration space of n-tuples in the complex plane as the space of monic squarefree polynomials, there is a natural equicritical stratification according to the multiplicities of the critical points. There is a lot to be interested in about these spaces: what are their fundamental groups (“stratified braid groups”)? Are they K(pi,1)’s? How much of the fundamental group is detected by the map back into the classical braid group? They are also amenable to study from a variety of viewpoints (most notably, they are related both to Hurwitz spaces and to spaces of meromorphic translation surface structures on the sphere). I will discuss some of my results thus far in this direction. Portions of this are joint with Peter Huxford.

April 9 - Reserve

April 11 - Reserve

April 16 - 

April 18 - 

April 23 - 

April 25 - 

May 2 - Amy Herron (University of Buffalo)

Triangle Presentations in ~A_2 Bruhat-Tits Buildings

The 1-skeleton of an ~A_2 Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from ”triangle presentations.” This abstract group either embeds into PGL(3, Fq((x))) or PGL(3, Qq), or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders q=2 or 3.  However, one abstract group that embeds into PGL(3,Fq((x))) for any prime power q is known via the trace function corresponding to the finite field of order q^3.  I have found a different method to derive this group via perfect difference sets.  This new method demonstrates a previously unknown connection between difference sets and ~A_2 buildings.  Moreover, this new method makes the final computation of triangle presentations easier, which is computationally valuable for large q.