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New York Group Theory Seminar: Friday, September 19, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Matthew Zaremsky (SUNY Albany)
Title: Aut(F_n) satisfies the Boone-Higman conjecture
Abstract:
The Boone-Higman conjecture (1973) predicts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. The "if" direction is true and easy, but the "only if" direction has been open for over 50 years. The conjecture is known to hold for various families of groups, perhaps most prominently the groups GL_n(Z) (due to work of Scott in 1984), and hyperbolic groups (due to work of Belk, Bleak, Matucci, and myself in 2023). In this talk I will discuss some recent work joint with Belk, Fournier-Facio, and Hyde establishing the conjecture for Aut(F_n), the group of automorphisms of the free group F_n, which has some surprisingly far-reaching consequences. If time permits I will also discuss some even more recent work joint with Fournier-Facio, P. Kropholler, and Lyman, in which we find roadblocks to our finitely presented simple groups being of type FP_\infty.
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New York Group Theory Seminar: Friday, September 26, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Benjamin Steinberg (City College of CUNY)
Title: Two-sided homological finiteness conditions and one-relator monoids
Abstract:
The homological finiteness conditions for groups have a natural generalization to rings, due to Kobayashi and Otto, which connects to Hochschild cohomology in much the same way the usual homological finiteness conditions relate to group homology. For monoids, the two-sided homological finiteness condition bi-FP_n implies both left and right-FP_n. Two-sided homological finiteness is better behaved than the one-sided conditions. Kobayashi proved that a monoid with a finite complete rewriting system is bi-FP_{\infty}, and so negating this property is a good way to show that a monoid does not admit a finite complete rewriting system.
Recently, the speaker and Bob Gray proved a conjecture of Kobayashi from 2000 that every one-relator monoid is left and right-FP_{\infty}. The question as to whether these monoids are bi-FP_{\infty} remains open. In this talk I'll discuss some idea of the proof that if the group of units of a special monoid is FP_n, then the monoid is bi-FP_n. From this it follows that special one relator monoids <X|w=1> are bi-FP_{\infty}. If time permits, I'll also indicate why the remaining incomporessible one-relator monoids are bi-FP_{\infty}, reducing us to the compressible case.
This is joint work with Bob Gray (University of East Anglia)
New York Group Theory Seminar: Friday, October 3, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Robbie Lyman (Rutgers Newark)
Title: Graphical models for topological groups
Abstract:
Recent breakthrough work of Rosendal allows for adapting the tools of geometric group theory to the study of any topological group. This is exciting in part because there is so much left to be done! The purpose of this talk is to give an account of my viewpoint on this story. Some of it represents joint work with Beth Branman, George Domat and Hannah Hoganson, where we introduce an analogue of the Cayley graph with respect to a finite generating set and apply it to the homeomorphism groups of countable ordinals.
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New York Group Theory Seminar: Friday, October 10, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Matthew Durham (Hunter College of CUNY)
Title: Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture
Abstract:
I will discuss recent work with Minsky and Sisto, in which we prove that mapping class groups of finite-type surfaces---and more generally, colorable hierarchically hyperbolic groups (HHGs)---are asymptotically CAT(0). This is a simple but powerful non-positive curvature property introduced by Kar, roughly requiring that the CAT(0) inequality holds up to sublinear error in the size of the triangle.
We use the asymptotically CAT(0) property to construct visual compactifications for colorable HHGs that provide Z-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit Z-structures. As an application, we prove that many HHGs satisfy the Farrell-Jones Conjecture, providing a new proof for mapping class groups (Bartels-Bestvina) and establishing the conjecture for extra-large type Artin groups.
To construct asymptotically CAT(0) metrics, we show that every colorable HHG admits a manifold-like family of local approximations by CAT(0) cube complexes, where transition maps are cubical almost-isomorphisms.
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New York Group Theory Seminar: Friday, October 17, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Be'eri Greenfeld (Hunter College of CUNY)
Title: Local Smallness, Global Largeness: An Invitation to the Burnside Problem(s)
Abstract:
If a group is finite, then all of its elements have finite order. In 1902, Burnside asked the converse: if a finitely generated group G has the property that every element has finite order, must G itself be finite? This is the general Burnside Problem, and more than sixty years later it was answered with a counterexample. Stronger variations of the problem lead to even deeper challenges. What if all elements have a uniformly bounded order? What if the group is assumed to be residually finite? These questions motivated major developments across several areas, including non-commutative rings, groups acting on trees, small cancellation theory, and non-associative algebra.
If time permits, we will discuss recent work (joint with Goffer and Olshanskii) on probabilistic versions of these problems, answering recent questions posed by Amir--Blachar--Gerasimova--Kozma. We construct finitely generated groups in which the behavior of element orders depends drastically on the choice of (standard) random walk: along one random walk, elements almost surely have uniformly bounded order, while along another, the group is almost surely torsion-free; and a group in which different random walks almost surely yield elements whose orders are pairwise coprime.
The talk will be mostly introductory, will not assume prior knowledge in the field, and is especially welcoming to students.
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New York Group Theory Seminar: Friday, October 31, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Lisa Carbone (Rutgers University)
Title: A group amalgam for the Monster Lie algebra
Abstract:
Recently, a group $G(\mathfrak m)$ given by generators and relations was associated to the Monster Lie algebra $\mathfrak m$ by Carbone, Jurisich and Murray. This was the first example of a Lie group analog for an infinite dimensional Borcherds algebra. However, a proof of non-triviality of the group $G(\mathfrak m)$ remained elusive. We discuss a different approach, where $G(\mathfrak m)$ has been constructed as a generalized amalgamated product, proving that the group $G(\mathfrak m)$ is non-trivial. We discuss the reasons why the classical approach to non-triviality failed and the role of AI in this work.
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New York Group Theory Seminar: Friday, November 7, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Siobhan O'Connor (CUNY Graduate Center)
Title: Potential Positivity in Free Groups
Abstract:
A freely reduced word in a free group F_r is called positive if it does not contain inverse generators. We say that a word is potentially positive if there is an automorphism of the free group that sends it to a positive word. In this talk I'll give a concrete introduction to automorphisms of the free group and present a nice necessary condition for a word in F_2 to be potentially positive. Then I'll present recent research results, such as a quadratic time algorithm for deciding whether or not a word in F_2 is potentially positive, the exact growth rate of potentially positive words of length n in F_2, and new lower bounds for the growth rates of potentially positive words in free groups of higher ranks. I'll also explain some implications of potential positivity in one-relator groups. Joint work with Lucy Koch-Hyde, Eamonn Olive, and Emma Dinowitz.
The talk will not assume prior knowledge in the field and is especially welcoming to students.
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New York Group Theory Seminar: Friday, November 14, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Alexander Ushakov (Stevens Institute of Technology)
Title: HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem
Abstract:
Let G be an HNN extension of a free group F with two equal associated normal subgroups H1=H2
of finite index. We prove that the word problem in G is decidable in polynomial time.
This result extends to the case where the subgroups H1=H2 are not normal, provided that the isomorphism
H1->H2 satisfies an additional condition.
Joint work with Hanwen Shen.
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New York Group Theory Seminar: Friday, November 21, 2025, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Alexei Miasnikov (Stevens Institute of Technology)
Title:
Abstract:
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