# Fall 2020 New York Group Theory Seminar

**During the Fall 2020 semester the New York Group Theory Seminar will meet on Thursdays, 4pm-5pm eastern time, via Zoom. Occasionally, talks may be scheduled at somewhat different times. Please check this page and the weekly seminar announcements for details. We hope that some in person NYGT talks may resume at the CUNY Graduate Center in the Spring 2021 semester. Stay safe, everyone! **

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***Thursday, September 10, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Yuri Gurevich** (University of Michigan).

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

Join Zoom Meeting

Meeting ID 981 0614 7266

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**: What, if anything, can be done in linear time?

Abstract:

The answer to the title question seems to be "Not much." Even sorting n items takes n x log(n) swaps. Actually, quite a bit can be done in linear time. In the first part of the talk we illustrate some known linear-time techniques (and pave the way to the second part).

Working on access control at Microsoft, we noticed that the most basic and useful access-control queries could be executed blazingly fast in practice. We wondered whether there was a theatrical foundation for the phenomenon. Eventually we came up with a logic calculus and an algorithm that, given a set of hypothesis and a set of queries, decides --- in linear time --- which of the queries follow from the hypotheses and which don't. In the second part of the talk, we explain how this is at all possible.

The presentation builds on joint work with Itay Neeman (UCLA Prof.), Carlos Cotrini and Ori Lahav (students at the time), and Artem Melentyev (a Microsoft intern at the time).

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A YouTube video of the talk is available below. A link to the video, which will open in a separate window, is also available here.

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***Thursday, September 17, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Abdul Zalloum** (Queen's University, Canada)

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

Join Zoom Meeting

Meeting ID: 972 4869 4024

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**: Regularity of Morse geodesics and growth of stable subgroups

Abstract:

The study of groups with ``hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by ``hyperbolic-like directions'', the notion of a *contracting *geodesic and that of a *Morse *geodesic. Since the property that every geodesic ray in metric space X is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. Generalizing work of Cannon, I will discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. This immediately implies that contracting geodesics can't exist in torsion groups. The Morse notion is a weaker notion than that of the contracting notion, in fact, building on work of Osin, Ol’shanskii, and Sapir, Fink gave an example of a torsion group which contains an infinite Morse geodesic. This seems to contradict the claim that Morse geodesics are "hyperbolic-like" directions. As an attempt to rectify this, Russell, Spriano, and Tran introduced a class of spaces where Morse (quasi)-geodesics satisfy some local-to-global property and they showed that many interesting examples live in such a class. In these spaces, Morse (quasi)-geodesics are expected to behave more reasonably like ``hyperbolic directions", therefore, such spaces/groups can be regarded as **good hosts **of Morse (quasi)-geodesics. I will discuss some continuation of their work where we show that in such spaces Morse geodesics form a regular language, give a characterization of stable subgroups in terms of regular languages. Time permitting, I will discuss few other applications of these automatic structures to the growth of stable subgroups and the dynamics of the action of such groups on their Morse boundaries. This work is joint with Cordes, Russell and Spriano."

A Youtube video of the talk is available below. A link to the vide is also available here (the link will open in a separate window).

Pdf slides of the talk are available here

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***Thursday, September 24, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Alexei Miasnikov (Stevens Institute of Technology)**

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

Join Zoom Meeting

Meeting ID/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**: The Diophantine problem in classical matrix groups

Abstract:

The Diophantine problem in a group (ring) G is decidable if there exists an algorithm that given a finite system of equations with coefficients in G decides whether or not the system has a solution in G. I will discuss the Diophantine problem in the groups the classical matrix groups G_n(R), where R is an associative unitary ring, n > 2, and G(n,R) is one of the groups GL(n,R), SL(n,R), T(n,R), UT(n,R), PGL(n,R), or PSL(n,R) (in the last two cases we assume that R is also commutative). The main result is that the Diophantine problem in G(n,R) with a chosen set of coefficients is Ptime reducible (Karp reducible) to the Diophantine problem in R with respect to a suitable set of coefficients in R. What is much more interesting is that the converse is also true. In the case when the ring R is finitely generated and commutative the result above allows one to clarify the situation completely, modulo a big conjecture in number theory. For not finitely generated rings, more so for uncountable rings, decidability of the Diophantine problem heavily depends on the set of constants. The case of classical fields of reals, complex and p-adic numbers is especially interesting, as well as the rings of p-adic integers (which is related to pro-p completions of groups). I am going to touch on this subject and, if time permits, show some surprising examples.

The talk is based on joint results with Mahmood Sohrabi.

A link to a YouTube video of the talk is available here (will open in a new window)

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***Thursday, October 1, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Laura Ciobanu (Heriott-Watt University)**

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

Meeting ID/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**: On computing fixed subgroups of endomorphisms in free groups

Abstract:

Given an endomorphism h of a free group F, the fixed subgroup of h consists of those elements x in F for which h(x)=x.

In this talk I will give some background on fixed subgroups in free groups, and then present an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank two.

This answers, for rank two, a question posed by Stallings in 1984 and a more recent question of Ventura. I will explain why general endomorphisms are more difficult than automorphisms, and in what ways our algorithm needs the restriction on the rank. This is joint work with Alan Logan.

A link to the YouTube video of the talk (will open in a new window)

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***Thursday, October 8, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Benjamin Steinberg (City College of CUNY)**

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

Meeting ID/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**: Simplicity of Nekrashevych algebras of contracting self-similar groups

Abstract:

A self-similar group is a group $G$ acting on the Cayley graph of a finitely generated free monoid $X^*$ (i.e., regular rooted tree) by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups like Grigorchuk's 2-group of intermediate growth are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk we discuss a recent result of the speaker and N. Szakacs (York/Szeged) characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like Gupta-Sidki groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

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***Thursday, October 15, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Annalisa Massaccesi (**Università degli Studi di Padova**)**

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

**Title**: Currents with coefficients in groups and applications to network optimization problems

Abstract:

In this seminar I will review the theory of flat G-chains, as they were introduced by H. W. Fleming in 1966, and currents with coefficients in groups. One of the most recent development of the theory concerns its application to the Steiner tree problem and other minimal network problems which are related with a Eulerian formulation of the branched optimal transport. Starting from a 2016 paper by A. Marchese and myself, I will show how these problems are equivalent to a mass-minimization problem in the framework of currents with coefficients in a (suitably chosen) normed group.

A link to the YouTube video of the talk (will open in a new window)

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*******Friday,**** October 23, 4pm-5pm (eastern time), New York Group Theory Seminar [****Note the Friday meeting time!****]**

**Speaker: Lisa Carbone** (Rutgers University, New Brunswick)

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

Join Zoom Meeting

**Title**: Constructing Lie group analogs for infinite dimensional Lie algebras

Abstract: We will describe several approaches to constructing analogs of Lie groups associated to infinite dimensional Lie algebras over fields and over Z. Our primary examples are Kac-Moody algebras and the monster Lie algebra which is an example of a Borcherds generalized Kac--Moody algebra.

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***Thursday, November 5, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Dawid Kielak (Oxford University)**

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

**Title**: Free-by-cyclic groups, polytopes, and algorithms

Abstract:

Using tricks from L^2 homology and some abstract algebra we will show

how to algorithmically compute the structure of the fibred cohomology

classes of free-by-cyclic groups and most 3-manifolds.

(Joint with Giles Gardam)

A link to the YouTube video of the talk (will open in a new window)

**Pdf slides of the talk**

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***Thursday, November 12, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Wenhao Wang (Vanderbilt University)**

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

**Title**: Dehn Functions of Finitely Presented Metabelian Groups

Abstract:

The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group $G$ is $2^{n^{2k}}$, where $k$ is the torsion-free rank of the abelianization $G_{ab}$, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.

A YouTube video of the talk (link will open in a new window)

Pdf slides of the talk

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***Thursday, November 19, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: George Domat (University of Utah)**

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

**Title**: Free products from spinning and rotating families

Abstract:

A natural goal of geometric group theory is to understand the algebraic properties of a group via geometry. The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup in a large ambient group. In particular, their work gives conditions under which the normal closure is a free product. I will talk about recent work that aims to unify their results and gives a significantly shorter proof of the theorem of DGO. This is joint work with M. Bestvina, R. Dickmann, S. Kwak, P. Patel, and E. Stark.

A link to the YouTube video of the talk (will open in a new window)

Pdf slides of the talk

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**Friday, December 4, 9am-2pm (eastern time)**

**Manhattan Algebra Day**** (one-day Zoom workshop)**

Meeting ID/password: If you did not receive the meeting password in an email workshop announcement message, please 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)

**Speakers and schedule****, all times U.S. eastern time:**

Morning Session

*9am: Anna Erschler (Ecole Normale Superieure, ENS de Paris)

Title: Ordering ratio function and Travelling Salesman breakpoint

for groups and metric spaces (joint with Ivan Mitrofanov)

A link to a YouTube video of the talk (will open in a new window)

*10am: Agatha Atkarskaya (Bar Ilan University and

Hebrew University of Jerusalem)

Title: Small cancellation rings

*Abstract*: The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. Joint results with A. Kanel-Belov, E. Plotkin, E. Rips.

A link to a YouTube video of the talk (will open in a new window)

*11am: Henry Wilton (University of Cambridge and Trinity College)

Title: Negative immersions and one-relator groups

*Abstract:* One-relator groups G=F/<<w>> pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints (eg Magnus’ theorem that the word problem is solvable). On the other hand, they are not susceptible to geometric techniques, since some of them (such as Baumslag—Solitar groups) exhibit extremely pathological behaviour.

I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder — the *primitivity rank* \pi(w), the smallest rank of a subgroup of F containing w as an imprimitive element. A sample application is that every subgroup of G of rank <\pi(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.

This is joint work with Larsen Louder

A link to a YouTube video of the talk (will open in a new window)

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*12noon: Lunch break

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Afternoon session

*1pm: Eugene Plotkin (Bar-Ilan University)

Title: On logical rigidity of groups

*Abstract:*

We will survey a series of recent developments in the area of .rst order descriptions

of linear groups. The goal is to illuminate the known results and to pose the new

problems relevant to logical characterizations of Chevalley groups and Kac-Moody

groups. We also dwell on the principal problem of isotipicity of .nitely generated

groups.

A link to a YouTube video of the talk (will open in a new window)

*2pm: Ualbai Umirbaev (Wayne State University)

Title: Automorphism groups of free algebras

*Abstract:* There are many interesting results on the structure of the automorphism

group Aut(F_n) and the outer automorphism group Out(F_n) of the free group F_n of rank

n. Unfortunately, the theory of automorphism groups of free algebras over a field is not

very rich and many problems are still open. I will describe some results and recall some

open questions on the structures of the automorphism groups of

(a) the polynomial algebra K[x_1; x_2; ... ,x_n] of rank n over a field K;

(b) the free associative algebra K<x_1; x_2, ..., x_n> of rank n over K; and

(c) the free Lie algebra Lie<x1; x2, ..., x_n> of rank n over K.

A link to a YouTube video of the talk (will open in a new window)

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In the Fall 2020 semester the New York Group Theory Seminar meets Thursday 4:00-5:00 p.m. U.S. eastern time via Zoom

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