# Fall 2022

# This semester, Fall 2022, the **New York Group Theory Seminar ** meets in a hybrid format with some in-person talks and some online talks. The seminar meets on Fridays. In -person talks are 4:15pm-5:15pm in room 6417 at the CUNY Graduate. Online Zoom talks are 4:00pm-5:00pm. All times are U.S. eastern time.

**New York Group Theory Seminar**meets in a hybrid format with some in-person talks and some online talks. The seminar meets on Fridays. In -person talks are 4:15pm-5:15pm in room 6417 at the CUNY Graduate. Online Zoom talks are 4:00pm-5:00pm. All times are U.S. eastern time.

**Graduate Center building access:** For the non-CUNY in-person seminar participants, please see the GUNY Graduate Center building access policy, Section "One-day visitors", at https://www.gc.cuny.edu/news/building-entry-policy

Such visitors will need to present a proof of vaccination or a recent negative COVID test when entering the Graduate Center building. The organizers need to e-mail the names of non-CUNY visitors to the GC security office at least 24 hours in advance. Therefore if you don't have a CUNY ID and want to attend a particular in-person meeting of the NYGT Seminar, please e-mail Ilya Kapovich at ik535@hunter.cuny.edu at least 24 hours in advance. Those non-CUNY participants who want to attend in-person meetings of the NYGT Seminar on a regular basis, can be added to the CUNY Cleared4 vaccine veificantion system, which would eliminate the need for the above case-by-case requests to the security office. If you would like to utilize this "recurring visitors" process, please e-mail ik535@hunter.cuny.edu as well for additional instructions. Participants with CUNY IDs can use their CUNY ID cards to enter all CUNY buildings.

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New York Group Theory Seminar

Friday, October 14, 2022, **4:****15*** pm* U.S. eastern time; room 6417 at the CUNY Graduate Center

**Speaker: **** Abid Ali**** (****Rutgers University****)**

**Title: ** Strong integrality of inversion subgroups of Kac-Moody groups

**Abstract: **

Let G be a Kac-Moody group over Q. This is a group associated to an infinite dimensional Lie algebra. The discrete form G(Z) of G has many appearances in physical applications. Mathematical descriptions of G(Z) is given by Tits. Tits’ functorial definition of these groups is not suitable for our purposes. We made some progress on finding a relationship between different discrete forms of G. Our results generalize Chevalley’s fundamental theorem on the integrality for finite dimensional semisimple Lie groups. We will report our joint work with Lisa Carbone (Rutgers), Dongwen Liu (Zhejiang) and Scott S Murray (Toronto).

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New York Group Theory Seminar

Friday, October 28, 2022, 4:15pm U.S. eastern time; room 6417 at the CUNY Graduate Center

**Speaker: ****Jean Pierre Mutangu****ha (IAS Princeton)**

**Title: **Canonical forms for free group automorphisms

**Abstract: **

The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

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New York Group Theory Seminar

Friday, November 4, 2022, 4:00pm U.S. eastern time

**Speaker:** Pavel Shumyatsky (University of Brasilia), online, via Zoom

If you are subscribed to the NYGT mailing list, you will receive the Zoom link for the talk via e-mail. If you are not subscribed, please e-mail Ilya Kapovich at ik535@hunter.cuny.edu for the Zoom link.

**Title: ****Commuting probability for subgroups of a finite group**

**Abstract: **

This is a joint work with Eloisa Detomi (University of Padova).\medskip

If $K$ is a subgroup of a finite group $G$, the probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. The probability that two randomly chosen elements of $G$ commute is denoted by $Pr(G)$. A well known theorem, due to P. M. Neumann, says that if $G$ is a finite group such that $Pr(G)\geq\epsilon$, then $G$ has a nilpotent normal subgroup $T$ of class at most $2$ such that both the index $[G:T]$ and the order $|[T,T]|$ are $\epsilon$-bounded.

In the talk we will discuss a stronger version of Neumann's theorem: if $K$ is a subgroup of $G$ such that $Pr(K,G)\geq\epsilon$, then there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded.

We will also discuss a number of corollaries of this result. A typical application is that if in the above theorem $K$ is the generalized Fitting subgroup $F^*(G)$, then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded.

A YouTube video of the talk is available here.

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New York Group Theory Seminar

Friday, November 11, 2022, 2:15pm U.S. eastern time; Room 4102 (Science Center) at the CUNY Graduate Center

(Please note the non-standard time/place of this talk.)

**Speaker: **** **** Anna Erschler (**École Normale Supérieure)

**Title: ** Poisson boundary of groups and Schreier graphs

**Abstract: **

In contrast with general Markov chains, random walks on groups enjoy stronger structural properties. We mention in particular characterisation of Varopoulos of recurrent random walks, the entropy criterion of Kaimanovich-Vershik and Derrienic for the triviality of the Poisson boundary. We explain old and recent results about related group invariants for groups.

Despite the fact that any regular graph can be realised as a Schreier graph and that many structural results fail for random walks on such graphs, we show that the properties of random walks on Schreier graphs can be related to and lead to new invariants of the group. We will discuss critical constant for recurrence; a joint work with V.Kaimanovich about random walks on the finitary symmetric groups and a related conjecture about locally finite groups; and a joint work with J.Frisch, N.Matte-Bon and T.Zheng about Poisson boundary of r.w. on Schreier graphs, including examples of Schreier graphs of Grigorchuk group. Such (infinitely supported) random walks show that the smallest example among possible infinite Schreier graphs, the ray, may admit measures with non-trivial boundary.

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New York Group Theory Seminar

Friday, November 18, 2022, 4:oopm U.S. eastern time, online, via Zoom

If you are subscribed to the NYGT mailing list, you will receive the Zoom link for the talk via e-mail. If you are not subscribed, please e-mail Ilya Kapovich at ik535@hunter.cuny.edu for the Zoom link.

**Speaker: **** **Agatha Atkarskaya (Einstein Institute of Mathematics)

**Title: ** Introduction to group-like small cancellation theory for rings

**Abstract: **

The structure of small cancellation groups is well known. Тhey are widely used in construction of groups with unusual properties (for example Burnside groups and Tarskii monster). We were interested in developing a similar theory for rings. However, such theory meets significant difficulties because, unlike groups, rings have two operations: addition and multiplication. I will speak about small cancellation conditions for rings that we introduced. These conditions provide the desired properties. I will discuss our way towards these conditions, examples and possible applications of small cancellation rings.

A YouTube video of the talk is available here.

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New York Group Theory Seminar

Friday, December 2, 2022, 4:15pm U.S. eastern time; room 6417 at the CUNY Graduate Center

**Speaker: **** ****Ramón Flores** ** (University of Seville)**

**Title: ** Graph-theoretic properties via right-angled Artin groups

**Abstract: **

In this talk we will discuss the natural bridge between Graph Theory and Group Theory which is provided by the structure of the right-angled Artin groups. After a general introduction to the subject, we will pay special attention to the identification of graph automorphisms, colorability and expansion. Joint work with Koberda and Kahrobaei.

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New York Group Theory Seminar

Friday, December 9, 2022, 4:00pm U.S. eastern time; online, via Zoom

If you are subscribed to the NYGT mailing list, you will receive the Zoom link for the talk via e-mail. If you are not subscribed, please e-mail Ilya Kapovich at ik535@hunter.cuny.edu for the Zoom link.

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

**Title: ** First order rigidity of homeomorphism groups of manifolds

**Abstract: **

I will discuss some aspects of the first order theory of homeomorphism groups of connected manifolds. The main result is as follows. Let M be a compact, connected manifold. There is a sentence S(M) in the language of groups such that if N is an arbitrary manifold and the homeomorphism group of N models S(M) then N is homeomorphic to M. This resolves a conjecture of Rubin from the 1980s. I will illustrate some of the ingredients of the proof, including an interpretation of second order arithmetic in the theory of homeomorphism groups of manifolds. This represents joint work with S. Kim and J. de la Nuez Gonzalez.

A YouTube video of the talk is available here.

# "*Commuting probability for subgroups of a finite group*," Pavel Shumyatsky (University of Brasilia), New York Group Theory Seminar, 11/04/2022

*Commuting probability for subgroups of a finite group*," Pavel Shumyatsky (University of Brasilia), New York Group Theory Seminar, 11/04/2022

# "Introduction to group-like small cancellation theory for rings", New York Group Theory Seminar, Agatha Atkarskaya (Hebrew University); 18/11/22

*First order rigidity of homeomorphism groups of manifolds*, Thomas Koberda (University of Virginia), NYGT Seminar, 9/12/2022

*First order rigidity of homeomorphism groups of manifolds*, Thomas Koberda (University of Virginia), NYGT Seminar, 9/12/2022