# Spring 2022

**Spring 2022 plans: **As the CUNY COVID building access policy is still uncertain, in the Spring 2022 semester the New York Group Theory Seminar will begin in the online format, on** Fridays at 4pm U.S. eastern time.** If the situation allows, we will try to schedule some in-person talks at the CUNY Graduate Center later during the semester.

**Video recordings of prior NYGT talks are available at the NYGT Youtube channel, ****https://www.youtube.com/channel/UCFI70J4QU4n_QuLwMjm4VIw**** **

**New York Group Theory Seminar**

**Friday, February 4, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Andrei Jaikin-Zapirain (Universidad Autónoma de Madrid)

**Title: **Free Q-groups are residually torsion-free nilpotent

**Abstract:**

I present a method that allows to prove that some groups are residually nilpotent. In particular, it can be applied to show that the free Q-groups are residually - (torsion-free nilpotent). This solves a conjecture of Baumslag.

A YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, February 1****8****, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Murray Elder (University of Technology Sydney)

**Title: **On groups presented by inverse-closed finite convergent length-reducing rewriting systems

**Abstract:**

In the 1980s Madlener and Otto asked for an algebraic characterisation of groups presented by finite, convergent, length-reducing rewriting systems, conjecturing that they are exactly the plain groups (free product of finitely many finite groups and infinite cyclic groups).

I will describe some recent results with Adam Piggott (ANU) on new geometric, algebraic and algorithmic properties of groups presented by (inverse-closed) finite, convergent, length-reducing rewriting systems.

A YouTube video of the talk is available here

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**New York Group Theory Seminar**

**Friday, February ****25****, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Matthew Zaremsky (SUNY Albany)

**Title: **Houghton-like groups from "shift-similar" groups

**Abstract:**

In joint work with Brendan Mallery, we introduce and study so called "shift-similar" groups. Self-similar groups are a well known class of groups, which in particular interact nicely with Higman-Thompson groups, and we introduce shift-similar groups as an analog that interacts nicely with Houghton groups. Shift-similar groups actually turn out to have many properties that self-similar groups do not, for example every finitely generated group embeds into some finitely generated shift-similar group, and there exist uncountably many finitely generated shift-similar groups. In this talk I will recall some background on self-similar groups, introduce shift-similar groups and the Houghton-like groups they produce, and discuss the aforementioned results plus some results about amenability. I will also highlight some open questions.

A YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, March ****18****, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Matt Brin (SUNY Binghamton)

**Title: **Ordinals and subgroups of Thompson’s group F

**Abstract:**

Subgroups of Thompson's group F can be very complex. We give a family of elementary amenable subgroups that models a large initial segment of countable ordinals. The family models not only the order structure but also the basic operations of sum, product and exponentiation with base little omega. Part of the appeal of the family is its ease of description. This is joint work with Collin Bleak and Justin Moore.

A YouTube video of the talk is available here.

**New York Group Theory Seminar**

**Friday, March ****25****, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Ilya Kapovich (Hunter College of CUNY)

**Title: **Primitivity rank for random elements in free groups

**Abstract:**

In 2014 Doron Puder introduced the notion of \emph{primitivity rank} $\pi(g)$ for a nontrivial element $g$ in a free group $F_r$ of rank $r$.

Namely, $\pi(g)$ is defined as the smallest rank of a subgroup $H$ of $F_r$ containing $g$ as a non-primitive element, or as $\infty$ if not such $H$ exists. The set of all subgroups $H$ of $F_r$ as above is denoted $Crit(g)$. It turned out that primitivity index of an element $w\in F_r$ is closely related to the questions about word-hyperbolicity and subgroup properties of the one-relator group $\langle F_r| w=1\rangle$.

We prove that if $r\ge 2$ and $F_2=F(x_1,\dots, x_r)$ is the free group of rank $r$, then, as $n\to\infty$, for a “random” element $w_n\in F_r$ of length $n$ with probability tending to $1$ one has $\pi(w)=r$ and $Crit(w)=\{F_r\}$. We discuss applications of this result to “word measures” on finite symmetric groups $S_N$, defined by such $w_n$.

A YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, ****April **** 1, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Thomas Koberda (University of Virginia)

**Title: **Hamiltonicity of graphs via right-angled Artin groups

**Abstract:**

I will discuss the dictionary between the algebraic structure of a right-angled Artin group and the combinatorics of the defining graph. I will then use the cohomology of a right-angled Artin group to provide a characterization of Hamiltonicity of the underlying graph. Along the way, I will describe some linear algebraic facts which appear to have been previously unknown. This talk represents joint with R. Flores and D. Kahrobaei.

A YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, April 22, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Marco Linton (University of Warwick)

**Title: **Hyperbolicity of certain one-relator groups

**Abstract:**

The primitivity rank of an element w of a free group F is defined as the minimal rank of a subgroup containing w as an imprimitive element. Recent work of Louder and Wilton has shown that there is a strong connection between this quantity and the subgroup structure of the one-relator group F/<<w>>. In particular, they show that one-relator groups whose defining relation has primitivity rank at least three cannot contain Baumslag—Solitar subgroups, leading them to conjecture that such groups are hyperbolic. In this talk, I will show how to confirm and strengthen this conjecture, providing some applications.

A YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, April ****29****, 2022, 4pm U.S. eastern time**

**Zoom talk** (Please e-mail ik535@hunter.cuny.edu for the Zoom link if you are not on the NYGT mailing list)

**Speaker: **Delaram Kahrobaei (City University of New York)

**Title: **Applied Group Theory in the Quantum and Artificial Intelligence Era

**Abstract:**

In this talk I present an overview of the current state-of-the-art in post-quantum group-based cryptography. I describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. I then describe some applications of combinatorial algebra in fully homomorphic encryption, and in particular homomorphic machine learning. In the end I will discuss several open problems in this direction. (https://arxiv.org/abs/2202.05917 accepted by the Notices of AMS).

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**New York Group Theory Seminar**

**Friday, ****May **** ****6****, 2022, 4pm U.S. eastern time**

**Zoom talk**: https://us02web.zoom.us/j/86199651429?pwd=RlcwYWttaHF3Zk9tU3BJNDBZMEsydz09

**Title: **Paul Schupp memorial session

**Abstract:**

Paul Schupp (March 12, 1937 - January 24, 2022) made a profound impact on the field of geometric group theory. This special meeting of the New York Group Theory Seminar will commemorate and honor Paul's work. There will be several short presentations by Paul's students and collaborators consisting of mathematical and personal reminiscences. An open mike period will follow.

A YouTube video of the Paul Schupp memorial NYGT Seminar is available here.

# "Free Q-groups are residually torsion-free nilpotent", Andrei Jaikin (Universidad Autónoma de Madrid), New York Group Theory Seminar, February 4, 2022

*On groups presented by inverse-closed finite convergent length-reducing rewriting systems, *Murray Elder (Sydney University of Technology), New York Group Theory Seminar, Friday, February 18, 2022

*On groups presented by inverse-closed finite convergent length-reducing rewriting systems,*Murray Elder (Sydney University of Technology), New York Group Theory Seminar, Friday, February 18, 2022

*Houghton-like groups from "shift-similar" groups*, Matthew Zaremsky (SUNY Albany), New York Group Theory Seminar, 02/25/22

*Houghton-like groups from "shift-similar" groups*, Matthew Zaremsky (SUNY Albany), New York Group Theory Seminar, 02/25/22

*Ordinals and subgroups of Thompson’s group F,* Matt Brin (Binghamton University), New York Group Theory Seminar, March 18, 2022

*Ordinals and subgroups of Thompson’s group F,*Matt Brin (Binghamton University), New York Group Theory Seminar, March 18, 2022

*Primitivity rank for random elements in free groups*, New York Group Theory Seminar, Ilya Kapovich (Hunter College), 03/25/2022

*Primitivity rank for random elements in free groups*, New York Group Theory Seminar, Ilya Kapovich (Hunter College), 03/25/2022

*Hamiltonicity of graphs via right-angled Artin groups*, Thomas Koberda (University of Virginia), New York Group Theory Seminar, 04/01/2022

*Hamiltonicity of graphs via right-angled Artin groups*, Thomas Koberda (University of Virginia), New York Group Theory Seminar, 04/01/2022

*Hyperbolicity of certain one-relator groups*, Marco Linton (University of Warwick), New York Group Theory Seminar, 04/22/2022

*Hyperbolicity of certain one-relator groups*, Marco Linton (University of Warwick), New York Group Theory Seminar, 04/22/2022

Paul Schupp memorial meeting of the New York Group Theory Seminar, May 6, 2022