# Fall 2021 New York Group Theory Seminar

**During the Fall 2021 semester the New York Group Theory Seminar will meet on ****Fridays****, 4pm-5pm eastern time, online ****via Zoom****. Occasionally, talks may be scheduled at somewhat different times. Please check this page and the weekly seminar announcements for details. **

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

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**New York Group Theory Seminar**

**Friday, October 1, 2021 4:00pm U.S. eastern time**

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

**The talk will be delivered ****via Zoom****. If you did not receive the meeting Zoom info from an NYGT mailing list message, please e-mail Ilya Kapovich at ****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: ****Cartan pairs of algebras**

**Abstract:**

In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting “nicely” inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).

In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the “Steinberg” algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its “Steinberg” algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.

It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.

This is joint work with Becky Armstrong, Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, Jacqui Ramagge and Aidan Sims

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

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**New York Group Theory Seminar**

**Friday, October ****8****, 2021 4:00pm U.S. eastern time**

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

**The talk will be delivered ****via Zoom****. If you did not receive the meeting Zoom info from an NYGT mailing list message, please e-mail Ilya Kapovich at ****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: ****On the Andrews-Curtis conjecture**

**Abstract:**

I am going to talk about the group-theoretic aspects of the Andrews-Curtis conjecture, some recent results, and some old.

From my viewpoint the Andrews-Curtis conjecture is not just a hard stand-alone question, coming from topology, but a host of very interesting problems in group theory.

**A link to the YouTube video of the talk is available here.**

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**New York Group Theory Seminar**

**Friday, ****October**** 15, 2021 4:00pm U.S. eastern time**

**Speaker: Chloe Papin (University of Rennes)**

**The talk will be delivered ****via Zoom****. If you did not receive the meeting Zoom info from an NYGT mailing list message, please e-mail Ilya Kapovich at ****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: ****A Whitehead algorithm for Generalized Baumslag-Solitar groups**

**Abstract:**

Baumslag-Solitar groups BS(p,q) =< a, t | ta^p t^{-1} = a^q > were first introduced as examples of non-Hopfian groups. They may be described using graphs of cyclic groups. In analogy with the study of Out(F_N) one can study their automorphisms through their action on an "outer space". After introducing generalized Baumslag-Solitar groups and their actions on trees, I will present an analogue of a Whitehead algorithm which takes an element of a free group and decides whether there exists a free factor which contains that element.

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

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**New York Group Theory Seminar**

**Friday, October 22, 2021, 4:00pm U.S. eastern time**

**Speaker: John Mackay (University of Bristol)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: **Conformal dimension and decompositions of hyperbolic groups

**Abstract:**

The boundary of a Gromov hyperbolic group carries a canonical family of metrics which determine the quasi-isometry type of the group. Pansu's conformal dimension of the boundary gives a natural and important quasi-isometric invariant. I will discuss how this invariant behaves when the group splits over two-ended subgroups (i.e. when the boundary has local cut points), and applications to the question of Bonk and Kleiner asking for a characterization of when this dimension equals one. Joint work with Matias Carrasco.

** A link to a YouTube video of the talk is available here.**

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**New York Group Theory Seminar**

**Friday, October 2****9****, 2021, 4:00pm U.S. eastern time**

**Speaker: ****Samuel J. **** ****Taylor**** ( Temple University)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: **Orientable maps and polynomial invariants of free-by-cyclic groups

**Abstract:**

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an *orientable* fully irreducible outer automorphism F and use it to characterize when the homological stretch factor of F is equal to its geometric stretch factor. This is joint work with Spencer Dowdall and Radhika Gupta.

**A link to a YouTube video of the talk is available here.**

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**New York Group Theory Seminar**

**Friday, November 5, 2021 4:00pm U.S. eastern time**

**Speaker: Alessandro Sisto (Heriot-Watt University)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: **A simple hierarchical hyperbolicity criterion and extra-large Artin groups

**Abstract:**

A hierarchically hyperbolic structure is some kind of coordinate system on a given metric spaces where the coordinates take values in hyperbolic spaces, and it gives a good understanding of the coarse geometry of the space. I will give a brief introduction to this notion and its consequences, discuss a simple criterion to show that a space or group is hierarchically hyperbolic, and illustrate an application of this criterion to the case of extra-large type Artin groups.

**A link to a YouTube video of the talk is available here.**

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**New York Group Theory Seminar**

**Friday, November 12, 2021 ****4:00pm U.S. eastern time**

**Speaker: ****Vadim Kaimanovich ****(University of ****Ottawa****)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: **Coincidence, equivalence and singularity of harmonic measures

**Abstract:**

In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures "invariant on average", with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of harmonic measures on the underlying step distributions on the group and discuss counterexamples related to the Minkowski and Denjoy measure classes on the boundary of the classical modular group. The talk is based on joint work with Behrang Forghani.

A link to a YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, November 19, 2021 4:00pm U.S. eastern time**

**Speaker: Alexander Ushakov (Stevens Institute of Technology)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: **Quadratic equations in Baumslag-Solitar groups

**Abstract:**

We prove that the Diophantine problem for quadratic equations in

unimodular and metabelian Baumslag-Solitar groups BS(m,n) is decidable

and belongs to NP. Furthermore, the problem is polynomial-time decidable if

|m|=|n|=1 and is NP-hard otherwise.

**A link to a YouTube video of the talk is available here.**

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**New York Group Theory Seminar**

**Friday, December 3, 2021 4:00pm U.S. eastern time**

**Speaker: Sarah Rees (University of Newcastle)**

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: The compressed word problem in relatively hyperbolic groups**

**Abstract:**

I'll discuss recent work with Derek Holt that proves that the compressed word

problem in groups that are hyperbolic relative to free abelian subgroups can

be solved in polynomial time. This result extends results of Lohrey, and of

Holt, Lohrey and Schleimer, for free groups and for word hyperbolic groups,

and our proof imitates the proofs of those results.

I'll define all the terms used in the title,

explain background that motivates the result, and outline the methods used in

the proof.

A link to a YouTube video of the talk is available here.

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**New York Group Theory Seminar**

**Friday, December ****10****, 2021 4:00pm U.S. eastern time**

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

**The talk will be delivered ****via Zoom****ik535@hunter.cuny.edu**** to a request for a Zoom link.**

**Title: Rich groups and weak second order logic**

**Abstract:**

“What can one describe by first-order formulas in a given group A?” - is an old and interesting question. Of course, this depends on the group A. For example, in a free group only cyclic subgroups (and the group itself) are definable in the first-order logic, but in a free monoid of finite rank any finitely generated submonoid is definable. A group A is called *rich *if the first-order logic in A is equivalent to the weak second order logic. Surprisingly, there are a lot of interesting groups, rings, semigroups, etc., which are rich. I will describe various algebraic, geometric, and algorithmic properties that are first-order definable in rich groups and apply these to some open problems. Weak second order logic can be introduced into algebraic structures in different ways: via HF-logic, or list superstructures over A, or computably enumerable infinite disjunctions and conjunctions, or via finite binary predicates, etc. I will describe a particular form of this logic which is especially convenient to use in algebra and show how to effectively translate such weak second order formulas into the equivalent first-order ones in the case of a rich group A.

**A link to a YouTube video of the talk is available here.**** **

In the Fall 2021 semester the New York Group Theory Seminar meets on Fridays 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 ik535@hunter.cuny.edu.

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

**Cartan pairs of algebras**, Ben Steinberg (City College of CUNY), NYGT Seminar, Friday, October 1, 2021

*, Ben Steinberg (City College of CUNY), NYGT Seminar, Friday, October 1, 2021*

**Cartan pairs of algebras****On the Andrews-Curtis conjecture**, Alexei Myasnikov (Stevens Institute of Technology), NYGT Seminar, Friday, October 8, 2021

*, Alexei Myasnikov (Stevens Institute of Technology), NYGT Seminar, Friday, October 8, 2021*

**On the Andrews-Curtis conjecture****A Whitehead algorithm for Generalized Baumslag-Solitar groups****, **Chloe Papin, NYGT Seminar, Friday, October 15, 2021

**Conformal dimension and decompositions of hyperbolic groups**, John Mackay, NYGT Seminar, Friday, October 22, 2021

*, John Mackay, NYGT Seminar, Friday, October 22, 2021*

**Conformal dimension and decompositions of hyperbolic groups****Orientable maps and polynomial invariants of free-by-cyclic groups**, Samuel Taylor (Temple University), New York Group Theory Seminar, 10/29/2021

*, Samuel Taylor (Temple University), New York Group Theory Seminar, 10/29/2021*

**Orientable maps and polynomial invariants of free-by-cyclic groups****A simple hierarchical hyperbolicity criterion and extra-large Artin groups**, Alex Sisto (Heriot-Watt University), New York Group Theory Seminar, November 5, 2021

*, Alex Sisto (Heriot-Watt University), New York Group Theory Seminar, November 5, 2021*

**A simple hierarchical hyperbolicity criterion and extra-large Artin groups****Coincidence, equivalence and singularity of harmonic measures**, Vadim Kaimanovich (University of Ottawa), New York Group Theory Seminar, November 12, 2021

*, Vadim Kaimanovich (University of Ottawa), New York Group Theory Seminar, November 12, 2021*

**Coincidence, equivalence and singularity of harmonic measures****Quadratic equations in Baumslag-Solitar groups**, Alexander Ushakov (Stevens Institute of Technology), New York Group Theory Seminar, 11/19/2021

*, Alexander Ushakov (Stevens Institute of Technology), New York Group Theory Seminar, 11/19/2021*

**Quadratic equations in Baumslag-Solitar groups****The compressed word problem in relatively hyperbolic groups**, Sarah Rees (University of Newcastle), New York Group Theory Seminar, December 3, 2021

*, Sarah Rees (University of Newcastle), New York Group Theory Seminar, December 3, 2021*

**The compressed word problem in relatively hyperbolic groups****Rich groups and weak second order logic**, Alexei Myasnikov (Stevens Institute of Technology), New York Group Theory Seminar, 12/10/2021

*, Alexei Myasnikov (Stevens Institute of Technology), New York Group Theory Seminar, 12/10/2021*

**Rich groups and weak second order logic**