Topology and Geometric Group Theory Seminars at the Ohio State University

Jan 13 -

Jan 15 - 

Jan 20 - Jan 22 - Special events: Zassenhaus Lecture series by June Huh 

Jan 27 - 

Jan 29 - 

Feb 3 - 

Feb 5 - Doron Puder (IAS, Tel Aviv)    

Joint speaker with the Combinatorics Seminar

 Invariants of words from random matrices

Let w be a word in a free group and let G be a finite (or more generally compact) group. A w-random element of G is obtained by substituting the letters of w with uniform random elements from G. For example, if w=xyxy^{-2}, the random element is ghgh^{-2}, with g and h independent uniformly random elements of G. Composing with linear representations of G, we get w-random matrices. A series of works over the last decade has revealed many intriguing phenomena around w-random elements in nice families of groups, such as the symmetric groups or the unitary groups. In particular, many invariants of words, some new and some well-known, play significant roles in this theory.

This story involves probability, combinatorics, topology, algebra and representation theory. In the talk, where all notions will be explained, I will try to give a flavor of this interesting theory.

Feb 10 - 

Feb 12 - Zhenghao Rao (Rutgers)

Immersed incompressible surfaces in hyperbolic manifolds

        The study of surface subgroups in 3-manifolds has drawn sustained attention for decades, motivated both by their intrinsic geometric richness and by their broad consequences in geometric topology, geometric group theory, and dynamics. A landmark result is the Surface Subgroup Theorem of Kahn–Markovic, which asserts that every cocompact Kleinian group contains a closed surface subgroup. In this talk, we will survey key developments in the subject and highlight our recent progress, including joint work with Jeremy Kahn and with Xiaolong Han and Jia Wan.

Feb 17 - 

Feb 19 - Katherine Goldman (McGill)

Injective metrics and affine hyperplane arrangements

A complex affine hyperplane arrangement is a locally finite collection of affine hyperplanes (complex codimension-1 subspaces) in a finite dimensional complex affine space. Since these subspaces have complex codimension 1, the complement of their union is a connected manifold. It is a broad, longstanding problem with many connections to different areas of mathematics to determine the arrangements for which this manifold is aspherical (has contractible universal cover). A subset of this problem dating back to the 1970s, commonly attributed to Arnol’d, Brieskorn, Thom, and Pham, concerns arrangements arising from reflection groups in real affine space.

One approach that has seen success is to construct a cell complex which is homotopy equivalent to this complement and endow it with some kind of ("singular") non-positive curvature. Along these lines, by showing that a specific cell complex (based on a construction of Falk) carries a so-called injective metric, we show that a broad class of affine arrangements (including the infinite families of affine reflection arrangements, modulo a conjecture about D_n-type) have aspherical complement. In particular, this provides some of the first examples of infinite affine arrangements which have aspherical complement, but do not arise from reflection groups. This is joint work with Jingyin Huang.

Feb 24 -  Inhyeok Choi  (KIAS)

Random subgraphs of the mapping class group

In percolation theory, one samples a random subgraph of a given graph by independently deleting each edge with probability 1-p. Benjamini and Schramm asked if a Cayley graph is nonamenable if and only if there exists a number p such that the random p-subgraph has infinitely many unbounded components. In this talk, I will explain an answer to this conjecture for the mapping class group and irreducible cubical groups. This is joint work with Donggyun Seo.

Feb 26 - Pratit Goswami (University of Oklahoma)

Dehn Functions of Coabelian Subgroups

The study of Dehn functions has developed into a major area of research in geometric group theory mainly because the growth types of these functions are quasi-isometry invariants of finitely presented groups. The Dehn function of a finitely presented group G is also connected to the complexity of solving the word problem in G namely, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation is recursive. In this talk, we will discuss new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. This is joint work with Noel Brady and Rob Merrell.

Mar 3 - Leslie Mavrakis (Utah)

Combinatorial Characterizations and Branched Manifolds

A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite number of triangulated n-balls such that every manifold in F has a triangulation that locally looks like one of these n-balls. In joint work with Daryl Cooper and Priyam Patel, we show that LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In current and future work, we use this equivalence to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD. In this talk, I will present the main ideas of the proof of the equivalence and if time permits, construct branched 3-manifolds for a few of the geometries.

Mar 5 - Xiangdong Xie (BGSU)

Pattern rigidity in nilpotent groups

A pattern in a group consists of the left cosets of a collection of subgroups. A quasi-isometry between two groups with patterns is pattern-preserving if there is a constant D such that the image of every left coset in the  domain pattern is at Hausdorff distance at most D from a left closet in the target pattern. There are two natural questions concerning PPQIs(pattern-preserving quasi-isometries). The first is to determine when there is a PPQI between two groups with patterns.  The second is whether PPQIs exhibit rigidity properties.  We will discuss the second question in the setting of nilpotent groups. We show that every  self PPQI is at finite distance from an automorphism if the subgroups intersect the center trivially and  generate the whole group and one of the following holds: (1) G is a 2-step   torsion free finitely generated nilpotent group;  (2) G is a 2-step simply connected nilpotent Lie group.

   This  is ongoing joint work with   Mitra Alizadeh, Hao Liang and Qingshan Zhou.

Mar 5 - Special event: Colloquium talk by Slava Krushkal  (University of Virginia)

The Andrews-Curtis conjecture and low-dimensional topology

       Mar 10 - John Etnyre (Georgia Tech.)

                                                                               Monoids, braids and the mapping class group. 

 I will discuss various ways to define submonoids of the braid group and the mapping class group based on various geometric structures. I will begin by recalling the well-known notions of quasipositive and strongly quasipositive braid and how they are related to complex and symplectic geometry. I will then discuss joint work with Baker and Van Horn-Morris that allows one to define other submonoids based on contact geometry. Many open questions will be discussed. 

Mar 12- Special event: Colloquium talk by John Etnyre (Georgia Tech.)

Mar 24 - 

Mar 26 - Gil Goffer (UCSD)

Invariant Random Subgroups

An invariant random subgroup is a probability measure on the space of closed subgroups of a locally compact group that is invariant under conjugation. Invariant random subgroups generalize normal subgroups and lattices, and arise naturally from measure-preserving group actions. In the talk, I will give an overview of the subject and present recent results on compact invariant random subgroups, joint with Cohen, Glöckner, and Lederle, and on ends of random subgroups, joint with Segev.

Mar 31 - Thang Nguyen (FSU)

Actions of lattices preserving flag structures

The general theme of this talk is understanding the possible actions of a group on closed manifolds. When the group is a lattice in a higher-rank semisimple Lie group, it is expected that all possible actions are combinations of those inherited from the Lie group itself. While this has been proved in a few cases, it remains largely open in general. We will examine a specific situation where the manifolds carry a flag structure and the actions preserve that structure, leading to a classification of such actions. This talk is based on joint work with Vincent Pecastaing.

April 2 - 

April 7 -  Corey Bregman (Tufts)

Diffeomorphism groups of reducible 3-manifolds

Geometrization is a powerful principle which has shaped our understanding of irreducible 3-manifolds and, more recently, the structure of their diffeomorphism groups. However, diffeomorphism groups of reducible 3-manifolds remain somewhat elusive.  In this talk we survey some recent results in this direction, using the action of the diffeomorphism group on a space of separating systems, a topological poset parametrizing all collections of embedded spheres which decompose a reducible 3-manifold into its irreducible factors. Our approach provides an effective tool for computation, which we exhibit through various applications. This is joint work with R. Boyd and J. Steinebrunner.

April 9-  Hyunki Min (UGa)

Tight contact structures and torus knots

One of the fundamental problems in contact topology is to classify contact structures on a given 3-manifold. In particular, classifying contact structures on surgeries along a given knot has been very poorly studied. The only fully understood case so far is that of the unknot (lens spaces); for all other knots we have only partial results, or none at all. Several topological and algebraic tools have been developed to attack this problem.

In this talk, we discuss recent developments and the strategy for classifying tight contact structures on surgeries along torus knots. This is joint work with John Etnyre, Bülent Tosun, and Konstantinos Varvarezos.

April 14 - Lizzy Teryoshin (UCSD)

April 16 - Yunhui Wu (Tsinghua University)

April 21 - Ishan Banerjee (OSU)

April 23 - Ishan Banerjee (OSU)

April 28 - Marit Bobb (MPI Leipzig)