In the Spring 2025 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 5417. The online Zoom talks will be on Fridays at 4:00pm U.S. eastern time.
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New York Group Theory Seminar: Friday, February 7, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Alexei Miasnikov (Stevens Institute of Technology)
Title: On open problems in group theory
Abstract:
I intend to discuss various open problems in group theory, highlighting both my longtime personal favorites and some newer ones. I’ll focus on what makes these problems captivating and explore the reasons behind the extensive research on some while others remain relatively unexplored. The main topics I’ll cover include equations in groups, model-theoretic problems, algorithmic problems, and one-relator groups. I’ll emphasize problems that aren’t isolated, unapproachable castles but rather serve as illustrations of current trends and reveal entirely new research directions (regrettably, some of them are both). Additionally, I’ll mention some classical group theory problems that would be intriguing to attempt to solve using AI and provide justification for my choice. The talk is not overly technical and is assessable for graduate students.
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New York Group Theory Seminar: Friday, February 21, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Alina Vdovina (City College of CUNY)
Title: New expanders and their applications
Abstract:
We present new expander coming from cube complexes. These expanders turn out to be useful in parametrized complexity applications
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New York Group Theory Seminar: Friday, March 7, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Alexander Ushakov (Stevens Institute of Technology)
Title: Spherical equations in $Z_2 \wr (F_2\times F_2)$ are undecidable
Abstract:
Recall that a standard form of a spherical equation over a group $G$ is
$$\prod_{j=1}^k z_j^{-1} c_j z_j=1\ \ \ \ (k\ge 1),$$
where $c_1, ... , c_k\in G$ are constants and $z_1, ... ,z_k$ are variables.
Define a set of spherical equations with $k$ conjugates
$$\Sph_k(G) =\Set{\prod_{j=1}^k z_i^{-1} c_i z_i = 1 : c_1,\dots,c_k\in G}$$
that also can be considered as the corresponding Diophantine problem (an algorithmic question to decide if a given equation has a solution or not). A mapping from $\Sph_{k}(G)$ to $\Sph_{k+1}(G)$ defined by
$\prod_{j=1}^k z_i^{-1} c_i z_i = 1 \mapsto \prod_{j=1}^k z_i^{-1} c_i z_i \cdot z_{k+1}^{-1}1z_{k+1}= 1$
is a linear-time reduction. Hence, slightly abusing notation, we write $\Sph_{k}(G) \subseteq \Sph_{k+1}(G)$.
This creates a hierarchy of spherical equations in which
\begin{itemize}
\item $\Sph_1(G)$ can be viewed as the set of word-equations;
\item $\Sph_2(G)$ can be viewed as the set of conjugacy-equations;
\item the set of all spherical equations is the union $\Sph(G)= \bigcup_{i=k}^\infty \Sph_k(G).$
\end{itemize}
Novikov and Boone constructed a finitely presented group $G$ with undecidable word problem ($\Sph_1(G)$). Miller constructed a finitely presented group $G$ with decidable word problem ($\Sph_1(G)$) and undecidable conjugacy problem ($\Sph_2(G)$). In this work we show that in $\Z_2 \wr (F_2\times F_2)$ the word problem and the conjugacy problems are decidable, but there exists $k\ge 2$ for which $\Sph_k(G)$ is undecidable.
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New York Group Theory Seminar: Friday, March 14, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Kasia Jankiewicz (University of California at Santa Cruz and IAS Princeton)
Title: Profinite properties of clean graphs of groups
Abstract:
A graph of groups is algebraically clean if the edge groups embedding in vertex groups are inclusions of free factors. I will discuss some profinite properties of such groups, and applications to Artin groups.
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New York Group Theory Seminar: Friday, March 21, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Catherine Pfaff (IAS Princeton and Queen's University)
Title: A cubist decomposition of the Handel-Mosher axis bundle & preferred geodesics in Outer space
Abstract:
Automorphisms of free groups are largely studied via their action on Culler-Vogtmann Outer space. However, unlike in hyperbolic space or Teichmuller space (surface) settings, the dynamically minimal (fully irreducible) free group automorphisms act on Culler-Vogtmann Outer space with a collection of axes, which Handel and Mosher call the axis bundle. Not much of the structure of this axis bundle has yet been understood. Together with Chi Cheuk Tsang, we prove that the axis bundle has a "cubist" structure and use this structure to find preferred axes for these automorphisms. This work can be seen as in analogy with that of Hamenstadt and Agol in the surface setting.
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New York Group Theory Seminar: Friday, April 4, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Siobhan O'Connor (CUNY Graduate Center)
Title: Orbit-blocking words and other problems about free groups
Abstract:
By strengthening known results about primitivity-blocking words in free groups, we show that for any nontrivial element w of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image of w. This has implications for the average-case complexity of Whitehead's problem.
This is joint work with Lucy Koch-Hyde, Eamonn Olive, and Vladimir Shpilrain.
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New York Group Theory Seminar: Friday, April 11, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Caglar Uyanik (University of Wisconsin - Madison)
Title: Lamination posets are invariants of free-by-cyclic groups
Abstract:
Every outer automorphism of a finite-rank free group F has a well-defined (possibly empty) set of "attracting laminations". In fact, these laminations are partially ordered by inclusion and thus give a canonical "lamination poset" associated to the automorphism. These structures were introduced by Bestvina-Feighn-Handel and play a crucial role in their proof of the Tits alternative for Out(F).
In this talk, I will explain that the poset of laminations is in fact an invariant of the associated free by cyclic group. That is, if a group can be expressed as a free-by-cyclic group in two ways, then the two outer automorphisms (of possibly different free groups) have order-isomorphic lamination posets. This is joint work with Spencer Dowdall, Yassine Guerch, Radhika Gupta, and Jean-Pierre Mutanguha.
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New York Group Theory Seminar: Friday, April 25, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Lev Shneerson( Hunter College of CUNY)
Title: On identities of inverse semigroups with zero defined by positive relators
Abstract:
In the 1960s, S.I. Adian initiated the study of abstract properties of monoids given by special presentations in which the right- hand sides or defining relations are empty. In particular, he established that up to isomorphism there are only two special monoids with nontrivial identities that are not groups. We consider a similar problem for the special Rees quotients of free inverse semigroup FIA over the alphabet A, i.e. inverse semigroups with zero given by presentations of the form S = Inv (A; Ci = 0 (i \in I)) where each Ci \in A+ is a positive word over the group alphabet A U A-1 and completely describe all cases when S satisfies nontrivial identities, including identities in signature with involution. We show that in each of these cases S is finitely presented in the class of inverse semigroups. We also give a new sufficient condition for which a finite set X of reduced words over the alphabet A U A-1 freely generates a free inverse subsemigroup of FIA and use it in our proofs.
This is a joint work with David Easdown (University of Sydney).
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New York Group Theory Seminar: Friday, May 2, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Michael Chapman (NYU)
Title: Property testing in groups
Abstract:
In theoretical computer science, property testing is the field which seeks algorithms that distinguish between objects that satisfy a certain property, and objects that are far from elements satisfying the property, by querying the object at a few random positions. We will describe several problems in group theory that fall into the property testing framework. Specifically, Group Stability in Permutations and Proper Subgroup Testing.
This talk is based on joint works with Oren Becker, Irit Dinur and Alex Lubotzky.
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New York Group Theory Seminar: Friday, May 9, 2025, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Eduardo Silva (University of Münster)
Title: Continuity of asymptotic entropy on groups
Abstract:
The asymptotic entropy of a random walk on a countable group is a non-negative number that determines the existence of non-constant bounded harmonic functions on the group. A natural question to ask is whether the asymptotic entropy, seen as a function of the step distribution of the random walk, is continuous. In this talk, I will explain two recent results on the continuity of asymptotic entropy: one for groups whose Poisson boundaries can be identified with a compact metric space carrying a unique stationary measure, and another for wreath products A ≀ Z^d, where A is a countable group and d ≥ 3.
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