Fall 2024
Fall 2024
In the Fall 2024 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, September 20, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Saul Schleimer (University of Warwick)
Title: Solving the word problem in the mapping class group in quasi-linear time
Abstract:
Mapping class groups of surfaces are of
fundamental importance in dynamics, geometric group theory, and
low-dimensional topology. The word problem for groups in general, the
definition of the mapping class group, its finite generation by
twists, and the solution to its word problem were all set out by Dehn
[1911, 1922, 1938]. Some of this material was rediscovered by
Lickorish [1960's] and then by Thurston [1970-80's] -- they gave
important applications of the mapping class group to the topology and
geometry of three-manifolds. In the past fifty years, various
mathematicians (including Penner, Mosher, Hamidi-Tehrani, Dylan
Thurston, Dynnikov) have given solutions to the word problem in the
mapping class group, using a variety of techniques. All of these
algorithms are quadratic-time.
We give an algorithm requiring only O(n log^3(n)) time. We do this by
combining Dynnikov's approach to curves on surfaces, M\"oller's
version of the half-GCD algorithm, and a delicate error analysis in
interval arithmetic.
This is joint work with Mark Bell.
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New York Group Theory Seminar: Friday, September 27, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Ilya Kapovich (Hunter College of CUNY)
Title: On two-generator subgroups of mapping torus groups
Abstract:
Motivated by the results of Jaco and Shalen about 3-manifold groups, we prove that if F is a free group (of possibly infinite rank), $\phi: F\to F$ is an injective endomorphism of $\phi$ and $G_\phi=\langle F,t| t x t^{-1} =\phi(x), x\in F\rangle$ is the mapping torus group of $\phi$ then every two-generator subgroup of $G_\phi$ is either free or a “sub-mapping torus.” For a fully irreducible automorphism $\phi$ of a finite rank free group $F_r$ this result implies that every two-generator subgroup of the free-by-cyclic group $G_\phi$ is either free, free abelian, a Klein bottle group or a subgroup of finite index in $G_\phi$; and if $\phi\in Out(F_r)$ is fully irreducible and atoroidal then every two-generator subgroup of $G_\phi$ is either free or of finite index in $G_\phi$. This talk is based on joint paper with Naomi Andrew, Edgar A. Bering IV and Stefano Vidussi, with an appendix by Peter Shalen.
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New York Group Theory Seminar: Friday, October 25, 2024, 4:15pm, room 5417, CUNY Graduate Center
Talk cancelled because of the speaker's illness
Speaker: Catherine Pfaff (IAS Princeton and Queen's University)
Title: Train track automata for outer automorphisms of free groups and geodesics in outer space
Abstract:
The outer automorphism group of the free group Out(F_r) acts as the isometry group on the deformation space of weighted graphs, Culler-Vogtmann Outer space CV_r. The train track theory of Bestvina-Feighn-Handel bridges studying topological representatives of the group elements and geodesics in this space it acts on. We use the asymptotic conjugacy class invariant of the Handel-Mosher ideal Whitehead graph to “stratify” the space of geodesics, and the dynamically minimal “fully irreducible” outer automorphisms, into train track automata for different ideal Whitehead graphs. We then also contextualize this work in the broader program of understanding the geodesic flow. While the flow in the closed hyperbolic manifold and Teichmuller space settings is ergodic, it is unclear whether graphs live in such a nice setting. We explain some of our indicators of certain properties the flow may have. Some results presented are joint with some combinations of Y. Algom-Kfir, D. Gagnier, I. Kapovich, J. Maher, L. Mosher, and S.J. Taylor.
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New York Group Theory Seminar: Friday, November 1, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Benjamin Steinberg (City College of CUNY)
Title: The homology of groupoids and groups associated to self-similar groups
Abstract:
In the mid 2000s, Nekrashevych associated to each self-similar group a C*-algebra, a group (called a Rover-Nekrashevych group) with simple commutator subgroup and a topological groupoid. Both the group and the C*-algebra are easily constructible from the groupoid. Using Rubin's theorem, Nekrashevych showed that the groupoid can be recovered from the group. There are natural algebraic invariants associated to each of these objects. One has the K-theory of the C*-algebra, the homology of the group and the homology of the groupoid. Xin Li showed that the rational homology of the groupoid determines the rational homology of the Rover-Nekrashevych group. If the groupoid is torsion-free, the homology determines the K-theory of the C*-algebra through a spectral sequence.
In this talk I discuss recent work with Alistair Miller where we construct an exact sequence that connects the homology of the Nekrashevych groupoid to that of the self-similar group via the virtual endomorphism and transfer maps. There is a similar exact sequence for the K-theory of the Nekrashevych C*-algebra. Using this exact sequence, we are able to explicitly compute the homology of the groupoids associated to the Grigorchuk group, the Grigorhcuk-Erschler group, Gupta-Sidki p-groups and many other famous examples of self-similar groups. It follows from our results and Li's results that the Rover-Nekrashevych groups associated to the aformentioned self-similar groups are rationally acyclic.
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New York Group Theory Seminar: Friday, November 8, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Alexei G Miasnikov (Stevens Institute of Technology)
Title: On QFA property of restricted wreath products of free abelian groups and their central
extensions
Abstract:
We show first that the restricted wreath product G of two free abelian groups of finite rank
is quasi-finitely axiomatizable (QFA), that is any finitely generated group elementarily equivalent to G is isomorphic to G. Then we prove that there are uncountably many
elementarily equivalent but pair-wise non-isomorphic finite central extensions of Z_2 by G.
This talk is based on a joint work with Olga Kharlampovich and Denis Osin.
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New York Group Theory Seminar: Friday, November 15, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Paige Hillen (UCSB)
Title: Latent symmetry of graphs and stretch factors in Out(F_n)
Abstract:
Given an irreducible element of Out(F_n), there is a graph and an irreducible "train track map" on this graph, which induces the outer automorphism on the fundamental group. The stretch factor of an outer automorphism measures the asymptotic growth rate of words in Fn under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of such an irreducible train track representative. I'll present work showing a lower bound for the stretch factor in terms of the edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of a graph G gives a lower bound on the number of folds required for any train track map on G. I'll use this to classify all single fold irreducible train track maps.
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New York Group Theory Seminar: Friday, November 22, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Francis Wagner (Ohio State University)
Title: Malnormal subgroups of finitely presented groups
Abstract:
Introduced by Sapir in the late 1990s, the `S-machine' is a computational model which resembles a multi-tape, non-deterministic Turing machine. This model was carefully conceived in order to be both computationally robust and interpretable as a multiple HNN extension of a free group. As such, S-machines have proved to be a remarkable tool in the study of groups. I will discuss a generalization of the S-machine which yields the following refinement of Higman's embedding theorem: Every finitely generated recursively presented group may be quasi-isometrically embedded as a malnormal subgroup of a finitely presented group; moreover, the decidability of the Word problem is preserved by this embedding.
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New York Group Theory Seminar: Friday, December 6, 2024, 4:15pm, room 5417, CUNY Graduate Center
Speaker: Doron Puder (Tel Aviv University and IAS Princeton)
Title: Word Measures on GL_n(q) and Free Group Algebras
Abstract:
Every word in a free group induces a measure on every finite group by substituting the letters of w by uniformly random elements of the group. The connection between word measures on the symmetric group and the poset of f.g. subgroups of the free group was established several years ago.
In a more recent project, joint with Danielle Ernst-West and Matan Seidel, we study word measures on matrix groups over finite fields. To our surprise, we discovered a beautiful connection with free group algebras, with interesting analogies with the case of the symmetric group. I will describe what we found, as well as some intriguing conjectures.
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