# Spring 2020

**All the talks are on ****Thursdays, 5:00pm-6:00pm in Room 5417**** at the CUNY Graduate Center.**

**Wine and cheese are served afterwards in the math lounge on the 4th floor (Room 42414)**

**Note: Due to the ongoing coronavirus crisis, as of March 12, 2020 in person meetings of the New York Group Theory Seminar are temporarily suspended.**

**We will experiment, for now on a case-by-case basis, with holding the NYGT Seminar meetings in the online webinar format. Please see more specific info in the calendar below. **

**----------------------------------------------------------------------------------------------------------------**

***February 6, 5pm-6pm, **Robert Young (New York University)

**Title**: Filling functions of arithmetic groups

Abstract:

Filling invariants measure the difficulty of filling a closed curve or sphere in a space with a ball. This is easy in nonpositively curved spaces, where spheres can be filled by geodesics, but it is more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. Gromov and Thurston conjectured that the difficulty of filling a sphere in such a lattice depends on the dimension of the sphere and the rank of the symmetric space. This conjecture has been proven in several special cases, and in this talk, I will describe the geometry behind the conjecture and sketch a proof of the general case. This is joint work with Enrico Leuzinger.

***February 13, 5:15pm-6;15pm, **Lee Mosher (Rutgers University - Newark)

**Title**: Stallings fold paths and problems of Nielsen and Whitehead

Abstract:

This is an expository talk on the rank n free group F_n, its automorphism group Aut(F_n), and its outer automorphism group Out(F_n). The object is to show how about how modern tools for studying these objects --- such the outer space of F_n and Stallings fold paths in outer space --- emerge naturally as tools for investigating some of the earliest problems in the study of free groups, problems which were originally formulated by Nielsen and Whitehead.

***February 20, 5pm-6pm, **Rostislav Grigorchuk (Texas A&M University)

**Title**: Some results on spectral theory of groups

Abstract:

Abstract: Given a marked group *(G,S)* (i.e. a group *G* and a generating set *S*) its spectrum is the spectrum of the discrete Laplacian on the Cayley graph *Γ(G,S)*.

Spectrum and associated spectral measures reflect various properties of the group, for instance amenability. Study of them and computation is difficult subject. Very little examples of computation of spectrum, as well as of the information about its shape and spectral type of the Laplacian are known.

In my talk, after a short introduction to the spectral theory of graphs and groups, I will present few recent results obtained for the groups of intermediate growth and for the lamplighter group.

In particular, I will explain why the answer to the question of A.Valette (1993) and K.Fujiwara (2016) "Can one hear the shape of a group?" is NO in a very strong sense. The presented results are based on a joint work with A.Dudko (intermediate growth case) and with B.Simanek (lamplighter).

***February 27, 5pm-6pm, **Christopher Natoli (CUNY Graduate Center)

**Title**: Non-∀-homogeneity in free groups

Abstract:

We study to which extent the first-order properties of an *n-*tuple $\bar a$ in a non-abelian free group determine its automorphic orbit. By the results of Perin-Sklinos and Ould Houcine, the free group is homogeneous, namely the first-order type of a tuple determines this tuple up to automorphism. We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not ∀-homogeneous (free group of rank 2 is ∀-homogeneous by the result of Nies). We also provide interesting examples of countable non-finitely generated groups elementary equivalent to free groups. These are joint results with O. Kharlampovich.

***March 5, 5pm-6pm, **Mahmood Sohrabi (Stevens Institute of Technology)

**Title**: First-order rigidity of some linear groups

Abstract:

Let *O* be the ring of integers of a number field, and *n>2* be a natural number. In this talk, I present our work on the study of bi-interpretability of the ring of integers Z with the special linear group SL(*n,O*), the general linear group GL(*n,O*), and the solvable group of all invertible uppertriangular matrices over O, T(*n,O*). For each of the groups mentioned above we provide a complete characterization of arbitrary models of their complete first-order theories, in particular we address their first-order rigidity. This is joint work with Alexei Myasnikov.

***March 12: No seminar**

-----------------------------------------------------

***March 19, 5pm-6pm (eastern time), New York Group Theory Seminar**

**Speaker: **Zoran Sunik (Hofstra University).

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

https://hofstra.zoom.us/j/385634430

Meeting ID: 385 634 430

**Title**: Deciding if a right-angled Artin group is free-by-free is NP-complete

Abstract:

We show that deciding if a right-angled Artin group is free-by-free is an NP-complete problem. The work is based on an earlier result by Susan Hermiller and the speaker stating that the right-angled Artin group AΓ defined by the graph Γ is free-by-free if and only if Γ is 2-breakable (a graph Γ is 2-breakable if there exists an independent set D of vertices in Γ such that every cycle in Γ contains as least two vertices from D). We reduce the 3SAT Problem to the problem of deciding if a given graph is 2-breakable (in fact, k-breakable, for any fixed k ≥ 1). Once it is shown that

the problem is NP-complete, it is not difficult to show that it stays NP-complete even if we restrict it to right-angled Artin groups defined by planar graphs. Note that the more special problem of deciding if a right-angled Artin group is free-by-infinite-cyclic has a very simple answer. Namely, it follows easily from known results that the following three statements are equivalent. (1) AΓ is free-by-infinite-cyclic. (2) Γ is a forest. (3) AΓ embeds in the right angled group defined by the path of length 3. (Joint work with David Carroll and Benjamin Francisco.)

**Please note: To participate in this webinar, you need to download and install the freely available basic version of Zoom ****https://zoom.us/download**

**A Youtube video of the talk is available here (clicking on this link will open a new window).**

**A pdf file of the slides of the talk ****is available here**

---------------------------------------------------------------

***March 26, 5pm-6pm (eastern time), New York Group Theory Seminar**

Speaker: Sergiy Merenkov (City College of CUNY)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

https://ccny.zoom.us/j/479396647

Meeting ID: 479 396 647

**Title**: Quasisymmetry group of the basilica Julia set

Abstract:

I plan to discuss how Thompson's group T acts on the basilica Julia set J (the Julia set of f(z)=z^2-1) by quasisymmetries, and show that the group generated by T and a certain inversion \iota (that interchanges the Fatou components that contain the critical cycle) is dense (in the uniform topology) in the group of all quasisymmetries of J, with uniform quasisymmetric distortion bounds. This is joint work with Misha Lyubich.

**Please note: To participate in this webinar, you need to download and install the freely available basic version of Zoom ****https://zoom.us/download**

-------------------------------------------------------------------------------------

***April 2, 4pm-5pm U.S. eastern time, New York Group Theory Seminar **

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

Meeting ID 686-861-660

Speaker: Alina Vdovina (University of Newcastle)

Title: Buildings, C*-algebras and new higher-dimensional analogues of the Thompson groups.

Abstract:

We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory.

The underlying building structure allows explicit computation of the K-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomophic groups.

**Please note: To participate in this webinar, you need to download and install the freely available basic version of Zoom ****https://zoom.us/download**

A Youtube video of the talk is available at this link

A pdf file with the talk slides is available here

----------------------------------------------------

***April 9, 5pm-6pm eastern time, New York Group Theory Seminar **

Speaker: Paul Schupp (University of Illinois at Urbana-Champaign)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

Meeting ID: 542 955 793

Password: If you did not receive the meeting password in the seminar announcement message, e-mail Ilya Kapovich at ik535@hunter.cuny.edu to request the password (please e-mail from a college/university e-mail account when making such a request).

**Title**: Coarse computabilty and the Hausdorff distance between Turing degrees. (How even more geometric group theory invaded the theory of computability.)

Abstract:

Joint work with Carl Jockusch.

Coarse computabilty studies how well arbitrary sets can be approximated in terms of computable sets. Define two sets *A* and *B* of natural numbers to be *coarsely similar*, written $A \thicksim_c B$,

if their symmetric difference $A \triangle B$ has density $0$ in the sense of classical asymptotic density

from number theory. This relation is an equivalence relation, so we consider

the space $\mathcal{S} = \mathcal{P}(\mathbb{N})/\thicksim_c $ of coarse similarity classes.

There is a natural density metric defined on $\mathcal{S}$ by

setting δ(*A*,*C*) to be the upper density of their symmetric difference.

The space $\mathcal{S}$ is very interesting. While neither separable nor compact, it is both complete and contractible. Indeed, $\mathcal{S}$ is a geodesic metric space so it is a

hyperbolic space in the sense of Gromov with the property that there are uncountably many different geodesics be any two

distinct points of $\mathcal{S}$.

Define the *core*, or* lower cone*, κ(**d**), of a Turing degree **d**

to be the family { [*A*] } of all classes of sets such that $A \le_T \boldsymbol{d}$.

The closure $\overline{\boldsymbol{d}}$ of the degree **d** is the

closure of κ(**d**) in $\mathcal{S}$. Define the distance *H*(**d**, **e**)

between two Turing degrees as the Hausdorff distance between their closures in $\mathcal{S}$.

This distance has an equivalent definition solely in terms of computability theory. It turns out that

the the Hausdorff distance between any two degrees is either 0,1/2 or 1.

A Youtube video of the talk is available at this link.

Pdf slides of the talk are available here.

**https://zoom.us/download**

--------------------------------------------

***April 16: 5pm-6pm eastern time, New York Group Theory Seminar **

Speaker: Tatiana Smirnova-Nagnibeda (University of Geneva)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

Meeting ID: 651 394 835

Password: If you did not receive the meeting password in the seminar announcement message, e-mail Ilya Kapovich at ik535@hunter.cuny.edu to request the password (please e-mail from a college/university e-mail account when making such a request).

Title: Spectra of Laplacians on Cayley and Schreier graphs.

Abstract: We are interested in Laplacians on graphs associated with finitely generated groups: Cayley graphs and, more generally, Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general.

We will discuss various techniques that allow to construct examples with different types of spectra -- connected, union of two intervals, totally disconnected -- and with various types of spectral measure. The problem of spectral rigidity will also be addressed.

A Youtube video of the talk is available at this link.

A pdf file of the talk's slides is available here.

-------------------------------------------------------------------------

***April 23, 5pm-6pm eastern time, New York Group Theory Seminar**

Speaker: Frank Wagner (Vanderbilt University),

**Seminar talk delivered remotely, as a ****Zoom webinar **

**Meeting ID: **452 756 4182

**https://vanderbilt.zoom.us/j/4527564182**

Password: If you did not receive the meeting password in the seminar announcement message, e-mail Ilya Kapovich at ik535@hunter.cuny.edu to request the password (please e-mail **from a college/university e-mail account** when making such a request). You may also request to be added to the NYGT Seminar mailing list, so that you can get weekly talk announcements, with all the necessary Zoom details, automatically.

**Title**: Torsion Subgroups of Groups with Cubic Dehn Function

Abstract:

The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that the same does not hold for finitely presented groups with Dehn function as small as cubic. In particular, for every $m \geq 2$ and sufficiently large odd integer $n$, there exists an embedding of the free Burnside group B(m,n) into a finitely presented group with cubic Dehn function.

A link to the Youtube video of the talk is available here.

A pdf file of the talk's slides is available here.

**-----------------------------------------------------------------------------------**

***April 30: 5pm-6pm eastern time, New York Group Theory Seminar **

Speaker: Alexander Hulpke (Colorado State University)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

https://illinois.zoom.us/j/190519482

Meeting ID: 190 519 482

Password: If you did not receive the meeting password in the seminar announcement message, e-mail Ilya Kapovich at ik535@hunter.cuny.edu to request the password (please e-mail **from a college/university e-mail account** when making such a request). You may also request to be added to the NYGT Seminar mailing list, so that you can get weekly talk announcements, with all the necessary Zoom details, automatically.

**Title**: Index computations in arithmetic groups

Abstract:

The question whether a subgroup, given by generators, has finite (and then which) index is a natural question in group theory. Unfortunately, for natural groups such as SL_n(Z) and SP_{2n}(Z), this question cannot have a general algorithmic solution. Nevertheless it is often possible to determine this information in many cases using a computer

I will describe some approaches to this problem and illustrate these in examples.

This is joint work with Alla Detinko (Hull) and Dane Flannery (Galway).

A link to the Youtube video of the talk is available here.

A pdf file of the talk's slides is posted here.

**------------------------------**

***May 7: 5pm-6pm eastern time, New York Group Theory Seminar **

Speaker: Emily Stark (University of Utah)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

Meeting ID: 696 447 067

https://huntercollege.zoom.us/j/696447067

Password: If you did not receive the meeting password in the seminar announcement message, e-mail Ilya Kapovich at ik535@hunter.cuny.edu to request the password (please e-mail from a college/university e-mail account when making such a request).

**Title: **Action rigidity for free products of hyperbolic manifold groups

Abstract:

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

A link to the Youtube video of the talk is available here.

A pdf file of the talk's slides is posted here.

_____________________________________________

***May 14: 5pm-6pm eastern time, New York Group Theory Seminar **

Speaker: Benson Farb (University of Chicago)

**Seminar talk delivered remotely, as a ****Zoom webinar **

Join Zoom Meeting

https://huntercollege.zoom.us/j/93092140466

Meeting ID: 930-9214-0466

**Title: **Mapping class groups of K3 surfaces from a Thurstonian point of view

Abstract:

The state of our understanding of homeomorphisms of 4-manifolds in 2020 is essentially that of our understanding of homeomorphisms of 2-manifolds in 1973, before Thurston changed everything. In this talk I will report on some first steps in a project (with Eduard Looijenga) whose ultimate goal is to change this.

The seminar meets Thursday 5:00-6:00 p.m. at the Graduate Center of the City University of New York (Room 5417).

The current organizers are:

Robert Gilman (Stevens Institute of Technology), rhgilman@gmail.com

Ilya Kapovich (Hunter College of CUNY), ikapovitch@gmail.com

Olga Kharlampovich (Hunter College of CUNY), okharlampovich@gmail.com,

Alexei Miasnikov (Stevens Institute of Technology), amiasnikov@gmail.com

Vladimir Shpilrain (City College of CUNY), shpilrain@yahoo.com

Benjamin Steinberg (City College of CUNY), bsteinberg@ccny.cuny.edu

If you would like to give a talk, or have a suggestion for a seminar speaker, please e-mail one of the organizers. If you want to be added to/removed from the NYGT Seminar mailing list, please e-mail Ilya Kapovich at ikapovitch@gmail.com.

You can also subscribe/unsubscribe for the NYGT mailing list directly, at

NYGT mailing list subscribe/unsubscribe page:

https://gc.listserv.cuny.edu/scripts/wa-gc.exe?SUBED1=NYGT&A=1