# Fall 2023

In the Fall 2023 semester the New York Group Theory Seminar will meet in a hybrid format, with most talks in-person and some talks online.

The in-person talks will be on Fridays at 4:15pm eastern time, room 6417. The online Zoom talks will be on Fridays at 4:00pm eastern time.

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New York Group Theory Seminar, Friday, September 29, 2023, 4:15pm, CUNY Graduate Center, room 6417

The talk was given over Zoom because of the weather emergency in NYC. A link to the Youtube video of the talk is available here.

Part II of the talk has been scheduled for Friday, October 6 (see below).

Speaker: Alexei G. Miasnikov (Stevens Institute of Technology)

Title: Musing on exponentiation in groups

Abstract:

Exponentiation in groups is an old and well-researched subject. The main theme here is to understand what a “non-commutative module” is in various classes of groups. Following Lyndon in 1994 V. Remeslennikov and myself introduced a notion of a group admitting exponentiation in an associative unitary ring R (now called R-groups). This is the most “freest and universal” exponentiation that works in all groups and it applies nicely to free and hyperbolic groups, free products with amalgamation and HNN extensions, etc. M. Amaglobeli started studying R-groups in varieties, in particular, nilpotent and solvable ones. However, if a group satisfies an identity the notion of exponentiation can be further adjusted to reflect more closely the nature of the group. Thus, in the class of nilpotent groups there is a famous P. Hall and A. Mal’cev’s exponentiation that gives a perfect notion of a “nilpotent non-commutative module”. Recently, working on first-order properties of free metabelian groups, we together with O. Kharlampovich explored an exponentiation that naturally occurs in metabelian groups. In this talk I will discuss all these exponentiations, the corresponding centroids and tensor completions, and how they relate to each other.

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New York Group Theory Seminar, Friday, October 6, 2023, 4:15pm, CUNY Graduate Center, room 6417

The talk will be given in person and partially Zoomcast: the speaker's slides will be shared over 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.

Speaker: Alexei G. Miasnikov (Stevens Institute of Technology)

Title: Musing on exponentiation in groups, part II

Abstract:

Exponentiation in groups is an old and well-researched subject. The main theme here is to understand what a “non-commutative module” is in various classes of groups. Following Lyndon in 1994 V. Remeslennikov and myself introduced a notion of a group admitting exponentiation in an associative unitary ring R (now called R-groups). This is the most “freest and universal” exponentiation that works in all groups and it applies nicely to free and hyperbolic groups, free products with amalgamation and HNN extensions, etc. M. Amaglobeli started studying R-groups in varieties, in particular, nilpotent and solvable ones. However, if a group satisfies an identity the notion of exponentiation can be further adjusted to reflect more closely the nature of the group. Thus, in the class of nilpotent groups there is a famous P. Hall and A. Mal’cev’s exponentiation that gives a perfect notion of a “nilpotent non-commutative module”. Recently, working on first-order properties of free metabelian groups, we together with O. Kharlampovich explored an exponentiation that naturally occurs in metabelian groups. In this talk I will discuss all these exponentiations, the corresponding centroids and tensor completions, and how they relate to each other.

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New York Group Theory Seminar, Friday, October 13, 2023, 4:15pm, CUNY Graduate Center, room 6417

The talk will be given in person and partially Zoomcast: the speaker's slides will be shared over 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.

Speaker: Ilya Kapovich (Hunter College)

Title: Primitivity index bounds in free groups, and the second Chebyshev function

Abstract:

Motivated by results about "untangling" closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups,

d_{prim}(g;F_N) and d_{simp}(g,F_N), where

1≠g∈F_N, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence d_{prim}(a^nb^n; F(a,b)) as

n→∞. Answering a question of Kapovich, we prove that this sequence is unbounded and that for n_i=lcm(1,2,...,i), we have

|d_{prim}(a^{n_i}b^{n_i}; F(a,b))-log(n_i)|=o(log(n_i)).

By contrast, we show that for all n≥2, one has

d_{simp}(a^nb^n;F(a,b))=2.

In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

The talk is based on a joint paper with Zachary Simon.

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

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New York Group Theory Seminar, Friday, October 20, 2023, 4:15pm, CUNY Graduate Center, room 6417

The talk will be given in person and partially Zoomcast: the speaker's slides will be shared over 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.

Speaker: Andrey Nikolaev (Stevens Institute of Technology)

Title: Nonstandard polynomials and nonstandard free group

Abstract:

Interpretation and bi-interpretation offer a novel approach to studying all structures elementarily equivalent to a given one. We use this approach to describe and study nonstandard models of the ring of polynomials, Laurent polynomials, and the group ring of a free group.

In presence of interpretation but not bi-interpretation, this approach produces a family of structures elementarily equivalent to a given one. We exploit this to introduce nonstandard models of a free group. As time permits, we discuss their main properties.

The talk is based on joint work with Alexei Miasnikov.

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

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New York Group Theory Seminar, Friday, October 27, 2023, 4:15pm, CUNY Graduate Center, room 6417

Speaker: Patricia Commins (University of Minnesota)

Title: Left regular bands with symmetry

Abstract:

Left regular bands (LRBs) are a special class of finite semigroups. They are often studied for their connections to Markov chains, but have many interesting properties in their own right, such as a rich connection to poset topology developed by Margolis, Saliola, and Steinberg. Many of the LRBs appearing in the literature are naturally equipped with group actions. In this talk, we will explore LRB semigroup algebras under these group actions, focusing on the structure of the invariant subalgebra and more generally, the semigroup algebra as a simultaneous representation of the group and the invariant subalgebra.

Our primary example will be the face monoid of the braid arrangement which is acted upon by the symmetric group. Bidigare proved the invariant subalgebra of the face semigroup algebra is (anti-)isomorphic to Solomon's descent algebra. We will explore the structure of the entire semigroup algebra as a simultaneous representation of the descent algebra and the symmetric group. If time permits, we may explore how this example extends to a more general class of LRBs.

No prior knowledge of LRBs or hyperplane arrangements will be assumed!

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

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New York Group Theory Seminar, Friday, November 10, 2023, 12:00pm U.S. eastern time, via Zoom (please note the non-standard time)

Zoom link: 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.

Speaker: Andrei Jaikin-Zapirain (Autonomous University of Madrid)

Title: Coherence of one-relator groups and their group algebras

Abstract:

In my talk, I will explain the main ideas of how to prove that one-relator groups and their group algebras over fields of characteristic zero are coherent. These results are based on joint work with Marco Linton.

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

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

[This will be a blackboard talk and it will not be available on Zoom]

Speaker: Daniel T. Wise (McGill University)

Title: An introduction to nonpositive immersions

Abstract:

A 2-complex X has nonpositive immersions if for every immersion Y to X with Y compact and connected, either euler(Y)\leq 0 or pi_1Y=1. In the first part of my talk, I will give a survey on nonpositive immersions and some variants definitions. And discuss the relationships with various properties of X and subgroup properties of pi_1X, culminating with the exciting recent developments of Linton and Jaikin-Zapirain.

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New York Group Theory Seminar, Friday, November 17, 2023, 1:00pm, CUNY Graduate Center, room 6496 (please note non-standard time and room)

Speaker: Anna Erschler (École Normale Supérieure)

Title: Liouville property for linear groups and group extensions

Abstract:

The talk is based on joint work with Josh Frisch and Mark Rychnovsky.

Poisson-Furstenberg boundary is a probability space defined by a random walk.

Its non-triviality is equivalent to the existence of non-bounded

harmonic functions.

Given a finitely generated group, the well-known Stability Problem

asks whether the non-triviality of the Poisson-Furstenberg boundary

depends on the choice of a simple random walk on the group. In a

joint work with Josh Frisch, we reduce the question of boundary

triviality on a linear group to induced random walks on appropriately

defined metabelian groups which we call metabelian blocks. In case of

linear groups over a field of characteristics p, we show that the

boundary of a simple random walk is non-trivial if and only if there

is a block containing a three dimensional wreath product as a

subgroup. In particular, the Stability Problem has a positive answer

in this class. In a work with Josh Frisch and Mark Rychnovsky, we

describe a criterion for boundary non-triviality for linear groups

over a field of characteristics 0. A new phenomenon, occurring only

in characteristics 0, is a possibility to admit as a block subgroup

certain metabelian groups, associated to multivariable polynomials

with a spaced polynomial property. We ask whether any irreducible

polynomial over Z has this property unless it is a generalised

cyclotomic one. In case of a positive answer to this question our

result would provide a characterization of linear groups which admit a finitely supported symmetric random walk with non-trivial boundary.

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

Speaker: Denis Osin (Vanderbilt University)

Title: Generic length functions on countable groups

Abstract:

The talk is based on joint work with A. Jarnevic and K.

Oyakawa. Let L(G) denote the space of integer-valued length functions

on a countable group G endowed with the topology of pointwise

convergence. Under mild assumptions on the group G, we prove that a

generic (in the Baire category sense) length function on G is a word

length and the associated Cayley graph is isomorphic to a certain

universal graph U independent of G. We also prove that generic length

functions on G are virtually indistinguishable from the

model-theoretic point of view. Topological transitivity of the action

of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

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New York Group Theory Seminar, Friday, December 8, 2023

Special one-day event: Complexity Day

Part I, online via Zoom, 9:00am-11:40am U.S. eastern time

If you do 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.

9:00-9:02 Welcoming remarks

9:02-9:05 Introduction

9:05 -9:35 Murray Elder (University of Technology Sydney)

Title: When is a Cayley graph geodetic?

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

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9:40-10:10 Enric Ventura (UPC Barcelona)

Title: The central tree property and some average case complexity results for algorithmic problems in free groups

Abstract: (joint work with Pascal Weil and Mallika Roy). We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude lower than the worst case complexity of the best known algorithms. This applies both to subgroups given by a fixed number of generators, and to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. Another application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity in a free group, whose average case complexity is a constant depending only on the rank of the ambient free group.

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

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10:15-10:45 Armin Weiss (University of Stuttgart)

Title: Parallel Computation for Word Problems in Groups: Baumslag-Solitar Groups and Related

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

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10:50-11:40 Markus Lohrey (University of Siegen)

Title: Streaming word problems

Abstract: We are interested in highly efficient algorithms for word problems of groups:

the algorithm should read the input word once from left to right symbol by symbol (such algorithms are known as streaming algorithms), spending ideally only constant time for each input letter. Moreover, the space used by the algorithm should be small, e.g. O(log n) if n is the length of the input word. To achieve these goals we need randomization: the algorithm is allowed to make random guesses and at the end it gives a correct answer (is the input word trivial in the underlying group?) with high probability. We show that for a large class of groups

such algorithms exist, where in particular the space complexity is bounded by O(log n). These groups are obtained by starting with finitely generated linear groups and closing up under the following operations: finite extensions, graph products, and wreath products where the left factor is f.g. abelian.

We also contrast this result with lower bounds. For instance, for Thompson’s group F every randomized streaming algorithm for the word problem of F has space complexity $\Omega(n)$ (n is again the length of the input word).

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

Afternoon break

Part II, in person, room 4102 (Science Center), CUNY Graduate Center, 1:30pm-5:00pm, U.S. eastern time

1:30pm-2:00pm Ilya Kapovich (Hunter College of CUNY)

Title: What we know and what we don't know about the generic-case complexity of Whitehead's Algorithm

2:10pm-2:40pm Alexei Myasnikov (Stevens Institute of Technology)

Title: Complexity, Cryptography, and ChatGPT

coffee break

3:00pm-3:30pm Vladimir Shpilrain (City College of CUNY)

Title: Complexity in SL_2(Z), Lyapunov exponent, and Cayley hash functions

3:40pm-4:10pm Alexander Ushakov (Stevens Institute of Technology)

Title: Spherical functions: hard on average and collision-free

Abstract: The goal of this work is to create bridges between problems of computational group theory and assumptions of the lattice-based cryptography.

We discuss spherical equations over some finite metabelian groups and functions defined by spherical equations. We show that properly designed spherical functions have good cryptographic properties and can be used to design hash functions (assuming that some lattice problems are hard).

4:20pm-4:50pm Robert Gilman (Stevens Institute of Technology)

Title: Negligible subsets of groups

Abstract: Generic-case complexity affords many insights into the behavior of group theoretic algorithms on a group G, but it is not clear that this complexity is independent of the chosen set of generators for G. We propose a solution.

wine and cheese

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