# Spring 2023

# In the Fall 2023, 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.

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

Effective January 25, 2023, non-CUNY visitors are no longer required to show a proof of COVID vaccination or a negative COVID test when entering the CUNY Graduate Center building. They will still need to show a photo ID when signing in at the security desk upon entering the building. CUNY faculty and students are still required to scan their Cleared4 app when entering the Graduate Center.

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New York Group Theory Seminar: Friday, February 3, 2023, 4:15pm, room 5382, CUNY Graduate Center

Speaker: Alexei Myasnikov (Stevens Institute of Technology)

Title: General algebraic schemes, non-standard groups, and first-order classification

Abstract:

In this talk I will discuss a new notion of an algebraic group scheme and the related class of “new” algebraic groups (which, of course, contains the classical ones). This leads to some interesting results on the first-order classification problems in groups and sheds new light on first-order rigidity and quasi finite axiomatization. The main focus will be on non-standard models of groups (aka non-standard analysis), especially on non-standard models of the finitely generated ones with decidable or recursively enumerable word problems. In particular, I will touch on non-standard free and hyperbolic groups, BS(1,k) groups, and, if time permits, Thompson groups.

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New York Group Theory Seminar: Friday, February 10, 2023, 4:15pm, room 5382, CUNY Graduate Center

Speaker: Olga Kharlampovich (Hunter College)

Title: Equations and first-order sentences in random groups.

Abstract:

We prove that a random group, in Gromov's density model with d<1/16 (such a random group is a small cancellation group) satisfies with overwhelming probability a universal-existential first-order sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group. This is based on our result that solutions of a system of equations in such a random group with overwhelming probability are images of solutions in a free group. We also discuss problems with higher densities that come from differences between hyperbolic and small cancellation groups. These are joint results with R. Sklinos.

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New York Group Theory Seminar: Friday, February 17, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Hadi Salmasian (University of Ottawa)

Title: Polynomial-type formulas for the minimal faithful dimension of finite p-groups

Abstract:

Given a finite group $G$, let $m(G)$ denote the smallest integer $n$ such that $G$ can be embedded in $GL(n,C)$. We show that for a broad class of $p$-groups $G$, the value of $m(G)$ is given by finitely many polynomial-type formulas. More precisely, we prove the following: let $\mathfrak g$ be a nilpotent Lie algebra over $\mathbb Z$. For any finite ring $R$, let $\mathfrak g_R$ denote the base extension of $\mathfrak g$ by $R$, and let $G_R$ denote the finite $p$-group associated to $\mathfrak g_R$ via the Lazard correspondence. We prove that if $R$ is a finite field of order $p^f$, the value of $m(G_R)$ is given by finitely many formulas of the form $fF(p^f)$ where $F(x)$ is a polynomial with non-negative integer coefficients. We prove analogous results for more general finite rings $R$ and compute $m(G_R)$ for several examples, including free nilpotent groups and free metabelian nilpotent groups. This talk is based on joint work with M. Bardestani, K. Mallahi-Karai, and D. Rumiantsau.

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New York Group Theory Seminar: Friday, February 24, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Catherine Pfaff (Queen's University)

Title: What happens when you iterate a free group automorphism & typical trees in the boundary of outer space

Abstract:

As with matrices and eigenvectors, one gains important information about a free group automorphism by studying limiting objects of its repeated iteration. At the same time, while matrices act on Euclidean, spherical, and hyperbolic spaces by symmetries, automorphisms of free groups act on the "deformation space" of metric graphs by symmetries, namely Culler-Vogtmann outer space. This beautiful interplay between a space and a group of symmetries yields a deeper understanding of both the space and symmetry group. We focus on how these limiting objects, groups, and spaces communicate with each other and describe "typical" objects. Results presented are joint work with I. Kapovich, J. Maher, and S.J. Taylor.

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New York Group Theory Seminar: Friday, March 3, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Eilidh McKemmie (Rutgers University)

Title: Galois groups of random additive polynomials

Abstract:

The Galois group of an additive polynomial over a finite field is contained in a finite general linear group. We will discuss three different probability distributions on these polynomials, and estimate the probability that a random additive polynomial has a "large" Galois group. Our computations use a trick that gives us characteristic polynomials of elements of the Galois group, so we may use our knowledge of the maximal subgroups of GL(n,q). This is joint work with Lior Bary-Soroker and Alexei Entin.

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New York Group Theory Seminar: Friday, March 10, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Alina Vdovina (City College of CUNY)

Title: Higher-dimensional automata, Ramanujan shifts and buildings

Abstract:

We suggest a new definition of higher-dimensional automata

motivated by cocompact quotients of buildings. We construct

infinite series of such automata and produce very explicit constructions of

Ramanujan higher-dimensional graphs.

The talk is based on joint results with Ievgen Bondarenko, Rostislav Grigorchuk

and Jakob Stix.

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New York Group Theory Seminar: Friday, March 31, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Damian Osajda (University of Wroclaw)

Title: Coxeter groups are biautomatic

Abstract:

I will present our, with Piotr Przytycki (McGill), recent proof of biautomaticity of Coxeter groups. This answers a long-standing open problem with previous positive results only in very special cases. From our construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings associated with Coxeter groups are biautomatic.

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New York Group Theory Seminar: Friday, April 14, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Sam Corson (University of the Basque Country)

Title: Many trivially generated groups are the same

Abstract:

When the algebraic generators of a group are all trivial then the group is also trivial. However, when the group is combinatorially described using infinite words, the “generators” can be trivial while the group is uncountable. I will give some background and present recent work showing that many natural examples of such groups are isomorphic.

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New York Group Theory Seminar: Friday, April 21, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Keino Brown (CUNY Graduate Center)

Title: Representation of limit groups and quantification of

subgroup separability.

Abstract:

We show that for any finitely generated non-abelian subgroup $H$ of a

limit group $L,$ there exists a finite-index subgroup $K$ containing

$H$ which is fully residually $H$. We also show that for any finitely

generated subgroup of a limit group, there is a finite-dimensional

representation of the limit group which separates the

subgroup in the induced Zariski topology. As a corollary, we establish

a polynomial upper bound on the size of the quotients used to separate a finitely generated

subgroup in a limit group. This generalizes the results about free and

surface groups by Lauder, McReynolds, and Patel (2017).

These are joint results with O. Kharlampovich.

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New York Group Theory Seminar: Friday, April 28, 2023, 4:15pm, room 6417, CUNY Graduate Center

Speaker: Julia Pevtsova (University of Washington)

Title: Small quantum group, tensor triangular geometry and the Springer resolution

Abstract:

Tensor triangular geometry (tt-geometry), formally introduced by Balmer in 2005, studies tensor triangulated categories by associating to them a geometric invariant, the (Zariski) spectrum of the category. After briefly introducing the general formalism of tt-geometry and some known examples coming from homotopy theory, algebraic geometry, and modular representation theory , I’ll focus on recent developments regarding the tt-spectrum of the representation category of a small quantum group. This is joint work with Cris Negron.

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New York Group Theory Seminar: Friday, May 5, 2023, 4:00pm, online, via Zoom

Zoom link:

https://us02web.zoom.us/j/83228210961?pwd=RnE2Ky9LRE15ODJ1MGF2UmRDZlIxUT09

Speaker: Benjamin Steinberg (City College of CUNY)

Title: Contractibility of the orbit space of Brown's p-subgroup complex - a new proof

Abstract:

K. Brown introduced in 1975 the $p$-subgroup complex of a finite group $G$. It is the simplicial complex whose vertices are the nontrivial $p$-subgroups of $G$, where a collection of subgroups spans a simplex if it is a chain. This complex was further studied by Quillen, who observed that for a finite group of Lie type $G$ with defining characteristic $p$, this complex is homotopy equivalent to the building of $G$. He also conjectured that the $p$-subgroup complex is contractible if and only if $G$ contains a nontrivial normal $p$-subgroup and proved his conjecture for solvable groups. The Quillen conjecture remains open but was proved for almost simple groups by Aschbacher and Kleidman, and strong reduction theorem was obtained by Aschbacher and Smith.

The group $G$ acts on its $p$-subgroups by conjugation and hence acts simplicially on the $p$-subgroup complex. Webb conjectured in 1987 that the orbit space of the $p$-subgroup complex is always contractible. He proved that its mod $p$ homology vanishes using methods from group cohomology. Webb's conjecture was first proved by Symonds in 98, and a number of other proofs have since appeared. All the proofs I am aware of go through Robinson's subcomplex, which is $G$-homotopy equivalent to Brown's. None of the proofs are explicit. Symonds computes the fundamental group and integral homology and uses the Hurewicz and Whitehead theorems. Bux gave an inductive proof using a variant of Bestvina-Brady style discrete Morse theory. In this talk, I will use Brown's theory of collapsing schemes to give an explicit sequence of elementary collapses that collapses the orbit space of Robinson's subcomplex to the vertex corresponding to the conjugacy class of $p$-Sylow subgroups.

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Friday, May 12, 2023

One-day special event in memory of Gilbert Baumslag (1933-2014) and Benjamin Fine (1948-2023)

All times are U.S. eastern time

Part 1, online, via Zoom

Zoom link: https://us02web.zoom.us/j/82410461381?pwd=YkE2dUdJUHNrTFJSSDJIY2pHN0t4UT09

9:00am Charles F, Miller (Melbourne University), "Collaborative mathematics"

9:30am Marco Linton (University of Oxford), "One-relator groups, virtually free-by-cyclic groups and coherence"

10:00am Martin Bridson (University of Oxford), "Sidki doubles, growth and Dehn functions"

11:00am Dennis Spellman (also on behalf of Anthony Gaglione), "Remembering Gilbert and Ben"

11:30am Gerhard Rosenberger (University of Hamburg), "A mathematical life with Ben Fine".

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12:00pm - 2:00pm Lunch break

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Part 2, in person, the Science Center (room 4102), CUNY Graduate Center

We plan to try to Zoomcast the afternoon in-person talks, if the technology cooperates. The Zoom link will be the same as for the morning talks:

https://us02web.zoom.us/j/82410461381?pwd=YkE2dUdJUHNrTFJSSDJIY2pHN0t4UT09

2:00pm Alexei Miasnikov (Stevens Institute of Technology), "Metabelian groups: isomorphism and first-order equivalence"

3:00pm Ilya Kapovich (Hunter College of CUNY), "Generic-case behavior of Whitehead's algorithm and geodesic currents on free groups"

4:00pm-4:15pm Wine and cheese break

4:15pm Robert Gilman (Stevens Institute of Technology), "Group Theoretic Cryptology"

4:45pm Vladimir Shpilrain (City College of CUNY), "On the problem of isomorphism to a particular group"

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