Topology and Geometric Group Theory Seminars at the Ohio State University

Aug 26 - 

Aug 28-

Sep  2 -

Sep 4-

Sep 9-

Sep 11-

Sep 16-  Yu-Chan Chang (OSU)

Dehn Functions of Bestvina--Brady Groups. 

Dehn functions are quasi-isometry invariants of finitely presented groups that measure the complexity of the word problem. Dison showed that the Dehn functions of Bestvina--Brady groups are at most quartic, and Brady gave examples of Bestvina--Brady groups that realize quadratic, cubic, and quartic Dehn functions.

In this talk, I will present a complete classification of the Dehn functions of Bestvina--Brady groups. This is joint work with Jerónimo García-Mejía and Matteo Migliorini. 

Sep 18-

Sep 23- Dragomir Saric (IAS, CUNY)

Infinite Riemann surfaces: the pants, the type and the quadratic differentials

An arbitrary infinite Riemann surface can be obtained by gluing infinitely many geodesic pairs of pants along cuffs. The Fenchel-Nielsen coordinates uniquely determine the Riemann surface structure. We give several sufficient conditions on the Fenchel-Nielsen coordinates to guarantee that the geodesic flow is ergodic. In the case when the glued pants have bounded cuff lengths, the condition can be expressed in terms of a simple random walk on a graph defined by the pants decomposition. 

We also find a surprising connection between the ergodicity of the geodesic flows and the horizontal foliations of (all) finite-area holomorphic quadratic differentials on the surface. This criterion is used to complement the sufficient conditions and to study function-theoretic properties of the surface.

Sep 25-

Sep 30- Thomas Ng (Brandeis)

Random quotients preserve negative curvature

Hyperbolic groups were introduced by Gromov in the 1980s and enjoy rich subgroup and quotient structure. Generalizations including relative, hierarchical, and acylindrical hyperbolicity, further highlight the deep connections between algebraic properties and metric negative curvature.  I will describe a model for constructing generic quotients of a group using independent random walks.  I will explain why such random quotients generically preserve the aforementioned aspects of negative curvature.  This is joint work with C. Abbott, D. Berlyne, G. Mangioni, and A. Rasmussen.

Oct 2- Hyein Choi (Rice)

Quasi-isometric embeddings of Ramanujan complexes.

Euclidean buildings (a.k.a. affine buildings and Bruhat-Tits buildings) are considered as a p-adic analogue of symmetric spaces. We show that there is no quasi-isometric embedding between the symmetric space of SL(n,R) and the Euclidean building of SL(n,Q_p). Generalizing this, we distinguish Ramanujan complexes constructed by Lubotzky-Samuels-Vishne as finite quotients of Euclidean buildings of PGL(n,F_p((y))) up to quasi-isometric embeddings. These complexes serve as high dimensional expanders with fruitful applications in mathematics and computer science.

Oct 7-

Oct 9- Franco Vargas Pallete (Arizona State)

                                                                Local minimization of fuzziness under high symmetry

For a given hyperbolic 3-manifold M with Fuchsian ends, it is a question of Canary and Taylor whether M minimizes the Hausdorff dimension of the limit set among its convex co-compact deformations. In this talk we will show how this holds locally if M is sufficiently symmetric. Based in work in progress with Sami Douba and Andrés Sambarino.

Oct 14- 

Oct 16- Autumn break

Oct 21- 

Oct 23-Mary He (University of Oklahoma)

Basmajian-type identities over non-Archimedean local fields

If S is a compact hyperbolic surface with geodesic boundary, Basmajian's identity expresses the length of the boundary as a sum over the orthogeodesics on the manifold. In this talk, we will prove Basmajian’s identity for projective Anosov representations \rho: \pi_1S \to PSL(d,k) where d \ge 2 and k is a non-Archimedean local field. Our identity exhibits a drastic difference from the ones over Archimedean fields. When d=2, we present a proof of the identity using Berkovic hyperbolic geometry. This is joint work with Chenxi Wu.

Oct 28-  Louisa Liles (OSU)  Lecture series 1

                                                              Witten-Reshetikhin-Turaev Invariants and Quantum Modular Forms

This talk is the first of a three-part series on quantum invariants of 3-manifolds. We will introduce Witten-Reshetikhin-Turaev (WRT) invariants, focusing on Lickorish's combinatorial description in terms of the Kauffman bracket. We will then discuss a pioneering q-series of Lawrence and Zagier, which not only unified the WRT invariants of the Poincaré Homology sphere, but was also a key first example Zagier's definition of a quantum modular form.

Oct 30- Louisa Liles (OSU) Lecture series 2

                                                                                           Lattice Homology and the Z-hat Invariant

This talk is the second of a three-part series on quantum invariants of 3-manifolds. We will give an introduction to negative-definite plumbed manifolds and two invariants associated to them. The first is a q-series given by Gukov, Pei, Putrov, and Vafa (GPPV) as an extension of Lawrence and Zagier's series. The GPPV invariant is known to unify the WRT invariants of all 3-manifolds on which it is defined, and is motivated by a theoretical physical framework predicting its existence for all 3-manifolds. The second invariant is Némethi’s theory of lattice homology, which builds on earlier work of Ozsváth and Szabó, and was recently showed by Zemke to be equivalent to Heegaard Floer homology. Time permitting, we will also discuss versions of these two invariants for knots inside three manifolds-- namely Gukov and Manolescu's two-variable series for knot complements, and knot lattice homology developed by Ozsváth, Stipsicz, and Szabó.

Nov 4- Louisa Liles (OSU)  Lecture series 3

                                                                                       (t,q)-Series Invariants of Seifert Manifolds

This is the third of a three-part lecture series on quantum invariants of 3-manifolds. Previously we discussed the GPPV q-series and Lattice homology. In this talk we will introduce a common refinement of these two invariants, which lives in a larger family of two-variable series developed by Ackhmechet, Johnson, and Krushkal. The speaker and Eleanor McSpirit provided the first known calculations of these new invariants for infinite families of manifolds-- first Brieskorn spheres, then Seifert fibered manifolds. We used these results to establish radial limits and quantum modularity properties whenever the t variable is fixed to be a root of unity. The resulting radial limits give rise to a novel family of $\zeta$-deformed WRT invariants, where $\zeta$ is any root of unity.

Nov 6-  Aleksander Skenderi (U. Wisconsin)  

Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups

An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equal the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan's property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3). 

In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup.

All concepts will be explained from scratch and we will focus on the case of matrix groups to make the presentation more concrete.

Nov 11- holiday

Nov 13- Amelia Pompilio (OSU)

Nov 18-Annette Karrer (OSU)

Nov 18-Ricky Lee (UCSB)

Nov 20- Jim Fowler

Nov 25- Elizabeth Buchanan (University of Iowa)

Nov 27- Thanksgiving

Dec 2-

Dec 4-

Dec 9- Qiuyu Ren (UC Berkeley)

Dec 11-