Fall 2023

Meeting dates: Tuesdays (Topology Seminar) 1:50-2:50 pm at MW 152 (or JR0387) and

Thursdays (Geometric Group Theory Seminar) from 1:50 to 2:50 pm at MW 152 (or SM2186)

Organizers - Jingyin Huang, Annette Karrer, Beibei Liu, Alex Margolis, Christoforos Neofytidis, Francis Wagner

The seminars will be run partially in person and partially virtually over Zoom for the Fall 2023 Semester

https://osu.zoom.us/j/95167217737?pwd=blExS2xLZWVGQWpQOXIvVnN2VGlZZz09

Meeting ID: 944 4945 1323 Password: 779408

August 29- Annette Karrer (OSU)

From Stallings' Theorem to connected components of Morse boundaries of graph of groups

Every finitely generated group G has an associated topological space, called a Morse boundary. It was introduced by a combination of Charney--Sultan and Cordes and captures the hyperbolic-like behavior of G at infinity. In this talk, I will motivate the study of Morse boundaries with Stallings' theorem. We will formulate a variant of Stallings' theorem for Gromov boundaries of Gromov-hyperbolic groups. As Morse boundaries generalize Gromov boundaries, this raises the question whether it is possible to formulate an analog for Morse boundaries. Motivated by this question, we will study connected components of Morse boundaries of graph of groups. We will focus on the case where the edge groups are undistorted and do not contribute to the Morse boundary of the ambient group. Results presented are joint with Elia Fioravanti.

August 31- Alex Margolis (OSU)

Model geometries dominated by locally finite graphs

The central theme of geometric group theory is to study groups via their actions on metric spaces. A model geometry of a finitely generated group is a proper geodesic metric space admitting a geometric group action. Every finitely generated group has a model geometry that is a locally finite graph, namely its Cayley graph with respect to a finite generating set. In this talk, I investigate which finitely generated groups G have the property that all model geometries of G are (essentially) locally finite graphs.

I introduce the notion of domination of metric spaces and give necessary and sufficient conditions for all model geometries of a finitely generated group to be dominated by a locally finite graph. This characterizes finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). Among groups of cohomological dimension two, the only such groups are surface groups and generalized Baumslag-Solitar groups.

Sep 5- Jingyin Huang (OSU)

Labeled four wheels and the K(pi,1) problem for reflection arrangement complements

The K(pi,1)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg MacLane spaces. We establish a close connection between a very simple property in metric graph theory about 4-cycles and the K(pi,1)-conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying the K(pi,1)-conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfies the K(pi,1)-conjecture.

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Sep 21- David Futer (Temple University)

Counting fixed points of pseudo-Anosov maps

Let S be a hyperbolic surface and f a pseudo-Anosov map on S. I will describe a result that predicts the number of fixed points of f, up to constants that depend only on the surface S. If f satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of f is coarsely equal to its translation length on the Teichmuller space of S. Without this mild condition, there is still a coarse formula.

This result and its proof has some applications to the search for surface subgroups of mapping class groups, to the hyperbolic volume of mapping tori, and to the knot Floer invariants of fibered hyperbolic knots. This is joint work with Tarik Aougab and Sam Taylor.

Sep 26- Jean Pierre Mutanguha (Princeton/IAS)

Canonical hierarchical decompositions of free-by-cyclic groups

Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus.

Sep 28- Sachin Gautam (OSU)

Monodromy representations of braid groups.

A general braid group is the fundamental group of a hyperplane complement modulo the action of a finite group. In this talk, I will give a brief survey of various differential equations with Fuchsian singularities coming from mathematical physics, and how their monodromy gives rise to representations of braid groups. I will also present the Drinfeld-Kohno theorem, and a few other results of similar vein, aimed at computing this monodromy using quantum groups.

Oct 3- Ishan Banejee (OSU)

Monodromy of algebraic curves in an algebraic surface

In this talk I will survey what is known about the monodromy group of all smooth algebraic curves embedded in an algebraic surface X of a fixed divisor class |D|. I will discuss how pi_1X can present obstructions to the monodromy group being finite index in the mapping class group. If time permits, I will discuss how the image of the monodromy group in quotients of the mapping class group by Johnson subgroups are 'as big as possible' assuming that D is sufficiently ample.

Oct 5- Stefanie Zbinden (Heriot-Watt University) (online-talk)

From strong contraction to hyperbolicity

For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

Oct 10- Guido Mislin (OSU/ETH)

Sidki doubles and conjectures in group theory

The Sidki double X(G) of a group G is obtained from the free product G ∗ G by making any element g from the first factor G commute with the corresponding element g in the second factor. There are natural surjections

G ∗ G ↠ X(G) ↠ G × G .

We present some results on the structure of Sidki Doubles and relate them to old conjectures in group theory. (Joint work with Indira Chatterji).

Oct 17- Federica Bertolotti (Pisa) - online talk

Triangulation Complexity vs. Integral Simplicial Volume: Mind the Gap

Triangulation complexity and Integral simplicial volume are two integral invariants that count the number of simplices needed to build a manifold. Triangulation complexity focuses on "nice" simplices, whereas integral simplicial volume allows for more degeneracy. Computing these two invariants is often challenging, and typically, we only have knowledge of their asymptotic behavior within infinite families of manifolds. Intriguingly, in many such investigations, these two invariants exhibit strikingly similar characteristics. This naturally prompts the question of whether they coincide, stay within a bounded distance from each other, or can have different asymptotic behavior. In this talk, we aim to explore and address this question. This is a joint work with Roberto Frigerio.

Oct 19- Hongbin Sun (Rutgers)

All 3-manifold groups are Grothendieck rigid

A finitely generated residually finite group G is said to be Grothendieck rigid if for any finitely generated proper subgroup H<G, the inclusion induced homomorphism \hat{H}\to\hat{G} on their profinite completions is not an isomorphism. There do exist finitely presented groups that are not Grothendieck rigid. We prove that all finitely generated 3-manifold groups are Grothendieck rigid. The proof relies on a precise description on non-separable subgroups of 3-manifold groups.

Oct 24- Francis Wagner (OSU)
A Mini-course on S-machines and their Applications

S-machines are an exceedingly useful tool in algorithmic group theory which facilitate the construction of finitely presented groups with desired (and sometimes surprising) properties. In this mini-course, we will provide an in-depth overview of this technical instrument.

In this first day, we discuss much of the background of the subject, beginning with an exposition on one interpretation of (multi-tape, non-deterministic) Turing machines. We relate this discussion to relevant topics in group theory such as the Word Problem of finitely generated groups and the Dehn function of finitely presented groups, then transition to a discussion of some major results in algorithmic group theory, in particular the Higman Embedding Theorem. All this serves as motivation for the invention of the S-machine, and so we conclude with a summary of their benefits, including a brief layout of results achieved through their means.

Oct 26- Francis Wagner (OSU)
A Mini-course on S-machines and their Applications

In this second day, we take a deep dive into the definition of S-machines, noting the key similarity and differences with the Turing machines discussed on the first day. This discussion culminates with the fundamental Simulation Theorem of Sapir, Birget, and Rips. Finally, we describe a different viewpoint on this machinery, reimagining the construction in a way that resembles an object-oriented programming language, providing better estimates and a refinement of the Simulation Theorem.

Oct 31- Francis Wagner (OSU)
A Mini-course on S-machines and their Applications

On this final day , we examine the finitely presented groups associated to an S-machine. We begin this discussion with an exposition on the fundamental tool of van Kampen diagrams, then use this to formalize the connection between the computational structure of the S-machine and the relational structure of the associated groups. We wrap up the mini-course by outlining the concept of graded small-cancellation theory and touching on a number of the consequential arguments underlying the results achieved through S-machines.

Nov 2- Mehdi Lejmi (CUNY)

Generalized almost-Kahler Ricci solitons.

In this talk, we introduce generalized almost-Kahler Ricci solitons metrics. In the Kahler Fano case, these metrics coincide with the Kahler Ricci solitons and in the symplectic Fano case they are obstructions to the existence of constant Chern scalar curvature almost-Kahler metrics. We discuss the deformations, the obstructions to the existence of such metrics and some examples.

Nov 7- Jonas Stelzig (LMU Munich) - online talk

Formality and dominant maps

A map of orientable equidimensional closed manifolds that induces a surjection in rational homology is called dominant. In general, one expects the target of a dominant map to be simpler than the source. A manifold is called formal if the graded-commutative

differential graded algebra of forms is quasi-isomorphic to the cohomology. This implies (under some condition on the fundamental

group) that the whole rational homotopy type is determined by the cohomology ring. I will explain that the target of a dominant map is

formal if the source is and discuss the context of this result. Joint work with Aleksandar Milivojevic and Leopold Zoller.

Nov 9- Jiming Ma (Fudan University)(online-talk)

Due to the time zone difference, this talk will be 9am in the morning.

Schwartz's complex hyperbolic surface

In this talk, we will study the topology of an infinite volume complex hyperbolic surface $M$, whose underlying topology is more complicated than plane bundles over real surfaces. R. Schwartz constructed $M$ in 2003, and Schwartz identified the 3-manifold at infinity of $M$. Now we may calculate the fundamental group of $M$, and more importantly, we may identify the topology of $M$.

Nov 9- Katherine Goldman (OSU)

CAT(0) and cubulated Shepherd groups

Shephard groups are generalizations of the complex reflection groups which act as symmetry groups of the so-called regular complex polytopes. In general, though, they may not be complex reflection groups; for example, their abstract presentation encompasses Coxeter groups, Artin groups, and graph products of cyclic groups. We extend a well known result that Coxeter groups are CAT(0) to a class of Shephard groups that have “enough” finite parabolic subgroups. We also show that in this setting, if the underlying Coxeter diagram is type FC, then the Shephard group is cubulated (i.e., acts properly and cocompactly on a CAT(0) cube complex). This provides many new non-trivial examples of CAT(0) and cubulated groups. Our method of proof combines the works of Charney-Davis on the Deligne complex for an Artin group and of Coxeter on the classification and properties of regular complex polytopes. Along the way we introduce a new criteria (based on work of Charney) for a simplicial complex made of simplices of shape A_3 to be CAT(1).

Nov 14- Tullia Dymarz (University of Wisconsin-Madison)

Tukia type theorems in quasi-isometric rigidity of SOL-like groups

The last step in Eskin-Fisher-Whyte's proof of quasi-isometric rigidity of lattices in the three dimensional solvable Lie group $SOL$ uses a theorem that states that certain groups of biLipschitz maps of $\R$ can be conjugated to act by similarities. We call such a theorem a Tukia-type theorem. This theorem is useful since $SOL$ is foliated by families of hyperbolic planes whose (parabolic) boundaries can be identified with $\R$ and quasi-isometries of $SOL$ induce biLipschitz maps of these boundaries. More generally a SOL-like group is a solvable Lie group that is foliated by negatively curved homogeneous spaces whose boundaries are nilpotent Lie groups. In recent work with Fisher-Xie we prove a Tukia-type theorem for a family of these nilpotent Lie groups called Carnot-by-Carnot groups. In this talk we will go through how Tukia-type theorems imply quasi-isometric rigidity and the challenges of proving such theorems for nilpotent Lie groups that are not $\R$.

Nov 16- Matthew Haulmark (Cornell)

From Separating Sets to Cube Complexes

Let M be a connected and locally path-connected metric space with no cut point, and suppose that G is a countable group acting on M by homeomorphisms. In this talk we will discuss how to get an action on a cube complex from a collection of separating sets in M that satisfies certain conditions. In the context of hyperbolic groups, our result provides an alternate route to the Sageev construction. This is joint work with Jason Manning.

Nov 28- Hoang Thanh Nguyen (FPT University) (online talk)

Due to the time zone difference, this talk will be 9am in the morning.

Property (QT) of 3-manifold groups

According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have the property (QT) if it can act isometrically on a finite product of quasi-trees, such that orbital maps are quasi-isometric embeddings. We show that all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition have property (QT). The main step of our proof is to establish property (QT) for the class of Croke-Kleiner admissible groups and relatively hyperbolic groups under natural assumptions. This is a joint work with Wenyuan Yang and Suzhen Han

Nov 28- Daren Chen (Caltech)

Bar-Natan homology for null-homologous links in RP^3 and genus bound

We will define a Bar-Natan homology for null homologous links in RP^3. As in the case for the usual Bar-Natan homology, this gives rise to a s-invariant and certain genus bound for null homologous knots in RP^3. More explicitly, it gives a genus bound for equivariant slice surface bounding the lift of the knot in S^3, such that the involution reverses the orientation on the surface.

Nov 30-Alexandra Edletzberger (Vienna) (online-talk)

Title: How to distinguish RACGs and RAAGs up to QI

Given a Right-Angled Coxeter group (RACG), we are interested in determining whether it is quasi-isometric (QI) to a Right-Angled Artin Group or not. In this talk we will introduce two tools that provide useful QI-invariants for this task: The Graph of Cylinders and the Maximal Product Region Graph. We will illustrate how they can distinguish RACGs from RAAGs up to QI by means of an example.

Dec 5- Koichi Oyakawa (Vanderbilt University)

Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Although group actions on hyperbolic spaces don’t always induce hyperfinite orbit equivalence relations on the Gromov boundary, some natural boundary actions were recently found to be hyperfinite. Examples of such actions include actions of hyperbolic groups and relatively hyperbolic groups on their Gromov boundary, actions of mapping class groups on arc graphs and curve graphs, and acylindrical group actions on trees. In this talk, I will show that any acylindrically hyperbolic group admits a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.

Dec 6- Mark Pengitore (University of Virginia)

Residual finiteness growth functions of surface groups with respect to characteristic quotients

Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.

Dec 7- Giuseppe Martone (Sam Houston State University,)

Title: Dynamics of cusped Hitchin representations

Abstract: Cusped Hitchin representations are generalizations of finite-area hyperbolic structures on a surface with cusps. They arise naturally in the context of higher rank Teichmüller theory, and, in this talk, we will focus on their dynamical properties. Specifically, in joint work with Bray, Canary, and Kao, we connect these surface group representations to the dynamics of countable Markov shifts and potentials with desirable (thermo)dynamical properties. Using these tools, we obtain orbit counting results for cusped Hitchin representations.