# Spring 2021 New York Group Theory Seminar

**During the Spring 2021 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. **

**If the COVID-19 situation changes, some NYGT talks may resume at the CUNY Graduate Center later during the Spring 2021 semester. Stay safe, everyone! **

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

**Speaker: Vladimir Shpilrain (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**: What, if anything, can be done in sublinear time?

Abstract:

In his talk on September 10, 2020, Yuri Gurevich discussed some algorithms that run in linear time (in the "length" of an input). We are going to take it up a notch and discuss what can be done in *sublinear *time; in particular, without reading the whole input but only a small part thereof. One well-known example is deciding divisibility of a decimal integer by 2, 5, or 10: this is done by reading just the last digit. We will discuss some less obvious examples from (semi)group theory.

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

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***Thursday, February 11, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Alina Vdovina (University of Newcastle)**

**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**: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

Abstract:

We will give a brief introduction to the theory of buildings and present their geometric, algebraic and arithmetic aspects. In particular, we present explicit constructions of infinite families of quaternionic cube complexes, covered by buildings. We will introduce new connections of geometric group theory and theoretical physics by using quaternionic lattices to find new infinite families of Drinfeld-Manin solutions of Yang-Baxter equations

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

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***Thursday, February 25, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Rizos Sklinos (Stevens Institute of Technology)**

**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**: Fields interpretable in the free group

Abstract:

We prove that no infinite field is interpretable in the first-order theory of non abelian free groups.

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

A pdf file with the slides of the talk

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

**Speaker: Tamar Bar-On (Bar-Ilan University)**

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

**Title**: Profinite completion of free profinite groups

Abstract:

The profinite completion of a free profinite group on infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question, which is equivalent to ask: what abstract embedding problems can a free profinite group solve.

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

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***Thursday, March 11, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Giles Gardam (University of Muenster)**

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

**Title**: Kaplansky's conjectures

Abstract:

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and finish with my recent counterexample to the unit conjecture.

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

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***Thursday, March 18, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Arman Darbinyan (Texas A&M)**

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

**Title**: Subgroups of left-orderable groups

Abstract:

A recent advancement in the theory of left-orderable groups

is the discovery of finitely generated left-orderable simple groups by

Hyde and Lodha. We will discuss a construction that extends this result

by showing that every countable left-orderable group is a subgroup of

such a group. In conjunction with this construction, we will also

discuss computability properties of left-orders in groups. Based on a

joint work with M. Steenbock.

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***Thursday, March 25, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Tullio Ceccherini-Silberstein (Università del Sannio)**

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

**Title**: Linear cellular automata, linear subshifts, and group rings

Abstract:

Let *G* be a group and let *V *be a finite dimensional vector space over a field *K*.

We equip *V*^{G} = {*x : G → V*} with the prodiscrete uniform structure,

the *G*-shift action ((*gx*)(*h*) := x(*g*^{-1}*h*)), and the natural structure of a *K*-vector space.

A *G*-invariant closed subspace *X* \leq *V*^{G} is called a linear subshift.

A linear subshift *X* ⊆ *V*^{G} is said to be of finite type provided that there exists a finite subset Ω ⊆ *G* and a subspace *W* ⊆ *V*^{Ω} such that

*X = X*(Ω,W) := {*x* ∈ *V*^{G}: (*gx*)|_{Ω} ∈ *W* for all *g* ∈ *G*}.

The group $G$ is said to be of **linear-Markov type** if for every finite dimensional vector space* V* over any field $K$, every linear subshift X ⊆ *V*^{G} is of finite type.

A uniformly continuous and *G*-equivariant *K*-linear map τ : *V*^{G} *→* *V*^{G} is called a **cellular automaton**.

The group *G* is said to be **linearly surjunctive** provided that for every finite dimensional vector *V*

space over any field *K* the following holds: every injective linear cellular automaton τ : *V*^{G} *→* *V*^{G} is surjective.

THEOREM 1 (CS-Coornaert 2007) Sofic groups are linearly surjunctive.

COROLLARY (Elek-Szabo 2004; CS-Coornaert 2007) Group rings of sofic groups are stably finite.

THEOREM 2 (CS-Coornaert-Phung 2020) A group is of linear-Markov type if and only if the group ring *K[G]*

is left-Noetherian for any field *K*.

COROLLARY 2 (CS-Coornaert-Phung 2020) Polycyclic-by-finite groups are of linearly-Markov type.

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***Thursday, April 15, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Tim Susse (Bard College at Simon's Rock)**

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

**Title**: Geometric Properties of Random Right-angled Coxeter Groups

Abstract:

Given a finite simplicial graph, we can generate a right-angled Coxeter group (RACG): each vertex corresponds to a generator and two generators commute if and only if the corresponding vertices are adjacent. This assignment is unique up to isomorphism, which allows us to characterize geometric and algebraic properties of the RACG using combinatorial properties of its generating graph. This also allows us to use a model of random graphs to study random RACGs.

In this talk we will focus on the divergence of a RACG. Using the Erdös-Renyi random graph model, we show that at a wide range of densities a random RACG asymptotically almost surely has quadratic divergence. We will also give a sharp threshold, below which a random RACG has (almost surely) at least cubic divergence, and above which its divergence is (almost surely) at most quadratic. This is joint work with Jason Behrstock, Victor Falgas-Ravry and Mark Hagen.

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

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***Thursday, April 22, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Olga Kharlampovich (Hunter College and the CUNY Graduate Center)**

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

**Title**: Universal theory of random groups

Abstract:

We will use Gromov's density model of randomness.

A random group at density d satisfies some property (of groups) P if the probability of occurrence of P tends to 1 as the length of relations goes to infinity.

Julia Knight conjectured that the limit of

the theories of random groups should converge to the theory of a free group.

We will show that this is true for the universal theory of a random group

at density d<1/16. Namely, every universal and every existential axiom of the free group is also true in a random group. Notice that a random group at density d<1/16 satisfies a small cancellation condition C'(1/8). We will also show that a random group at density d<1/2 is not a limit group (for a few relations model this was proved by Ho when the number of generators is less than the number of relations). These are joint results with R. Sklinos.

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***Thursday, April 29, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Ilya Kapovich (Hunter College and the CUNY Graduate Center)**

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

**Title**: Nonlinear words and free groups

Abstract:

An important theme in the study of combinatorics of words involves looking for models of *nonlinear* words, that is words that are not indexed by segments of integers. We discuss one such model arising from the theory of Stallings subgroup graphs. This model naturally leads to the notion of *subset currents* on free groups (and on other word-hyperbolic groups) which are measure-theoretic analogs of conjugacy classes of finitely generated subgroups. Many new features manifest themselves in this context, including connections with the Hanna Neumann Conjecture and Whitehead's algorithm for subgroups.

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

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***Thursday, May 6, 2021, 4pm-5pm (eastern time), New York Group Theory Seminar**

**Speaker: Benjamin Fine (Fairfield University)**

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

**Title**: Elementary and universal theories of group rings

Abstract:

Joint work with Anthony Gaglione, Martin Kreuzer, Gerhard Rosenberger and Dennis Spellman

In a series of papers the above authors examined the relationship between the universal and elementary theory of a group ring *R*[*G*] and the corresponding universal and elementary theory of the associated group *G* and ring *R*. Here we assume that *R* is a commutative ring with identity 1 ≠ 0. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided. Kharlampovich and Myasnikov as part of the proof of the Tarskii theorems prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. We examined and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if *R*[*G*] is elementarily equivalent to *S*[*H*] then simultaneously the group *G* is elementarily equivalent to the group *H * and the ring *R* is elementarily equivalent to the ring *S* with respect to the appropriate languages. Further if *G* is universally equivalent to a nonabelian free group *F* and *R* is universally equivalent to the integers $\mathbb{Z}$ then $R[G]$ is universally equivalent to $\mathbb{Z}[F]$ again with

respect to an appropriate language. It was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring

$R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$.

The structure of group rings is related to the **Kaplansky zero-divisor conjecture**. A **Kaplansky group** is a torsion-free gorup which satisfies the Kaplansky conjecture. We next show that each of the classes of left-orderable groups and orderable

groups is a quasivariety with undecidable theory. In the case of orderable

groups, we find an explicit set of universal axioms. We have that $\cal{K}$ the class of Kaplansky groups is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in $\cal{K}$ or more generally two torsion-free groups are universally equivalent.

Finally we consider *F* to be a rank 2 free group and $\mathbb{Z}$ be the ring of integers. we call $\Z[F]$ a free group ring. Examining the universal theory of the free group ring $\Z[F]$ the hazy conjecture was made that the universal sentences true in $\Z[F]$ are precisely the universal sentences true in *F* modified appropriately for group ring theory and the converse that the universal sentences true in *F* are the universal sentences true in $\Z[F]$ modified appropriately for group theory. We prove that this conjecture is true in terms of axiom systems for $\Z[F]$.

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

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In the Spring 2021 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), ik535@hunter.cuny.edu,

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.

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NYGT mailing list subscribe/unsubscribe page:

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