Links to seminar schedule for previous years: 2010/11, 2011/12, 2012/13, 2014/15, Spring 2016, Fall 2016, Spring 2017, 2017/18, 2018/19 2019/20 Fall 2020 Spring 2021
Organizers: Spencer Dowdall & Denis Osin
Time: Wednesdays, 4:10–5:00pm US Central Time
Location: SC 1312
Format: The seminar will typically meet in-person. However, the seminar will occasionally meet Online via Zoom Meeting as the situation merits; such as for outside speakers who are not able to travel to Vanderbilt. Please email one of the organizers for the Zoom Meeting invitation and join link.
Mailing List: If you would like to be added the seminar mailing list and receive regular announcments, please contact an organizer.
Samuel Shepherd (Vanderbilt)
Title: A version of omnipotence for virtually special cubulated groups
Abstract: The theory of group actions on CAT(0) cube complexes is a rich and successful area of research within geometric group theory, particularly regarding the class of virtually special cubulated groups. I will give some background on this and explain my new result for such groups, which allows you to control the orders of the images of certain collections of elements when mapping to a finite group.
Srivatsav Kunnawalkam Elayavalli (Vanderbilt)
Title: Proper proximality for groups acting on trees
Abstract: I will discuss joint work with Changying Ding, where we show that many groups acting on trees are properly proximal in the sense of Boutonnet-Ioana-Peterson.
Andrew Jarnevic (Vanderbilt)
Title: The Geometry of Subgroup Embeddings and Asymptotic Cones
Abstract: Given a finitely generated subgroup \(H\) of a finitely generated group \(G\) and a non-principal ultrafilter \(\omega\), we consider a natural subspace, \(Cone^{\omega}_{G}(H)\), of the asymptotic cone of \(G\), \(Cone^{\omega}(G)\), corresponding to \(H\). We discuss how the connectedness and convexity of \(Cone^{\omega}_{G}(H)\) are related to natural properties of the embedding of \(H\) in \(G\). We begin by defining a generalization of the distortion function which determines whether \(Cone^{\omega}_{G}(H)\) is connected. We then show that whether \(H\) is strongly quasi-convex in \(G\) is detected by a natural convexity property of \(Cone^{\omega}_{G}(H)\) in \(Cone^{\omega}(G)\).
Jiawei Han (Vanderbilt)
Title: Growth Of Dehn Twist and Pseudo-Anosov Conjugacy Classes in Teichmüller Space
Abstract: Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichmüller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. In contrast, we first show the number of Dehn twist lattice points intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{hR/2}$. Moreover, we show the number of all multi-twists lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R e^{hR/2}$. Furthermore, we show for any pseudo-Anosov mapping class $f$, there exists a power $n$, such that the number of lattice points of the $f^n$ conjugacy class intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{hR/2}$. Finally, we could discuss a few open questions and a conjecture. The main results in this talk are from author's papers arXiv:2105.08624 and arXiv:2105.08640.
Sergio Zamora Barrera (Penn State)
Title: The global shape of universal covers
Abstract: If one starts with the universal cover of a compact space, and looks at it from very far, what would the limiting shape be? It is well known (Gromov-Pansu Theorem) that if there is a limiting shape, then it must be a Carnot-Carathéodory group; a simply connected nilpotent Lie group with a special type of invariant metric.
I will talk about the similar problem of studying the limit shapes of universal covers of
sequences of spaces shrinking to a point.
Bogdan Chornomaz (Vanderbilt)
Title: S-Machines can emulate Turing machines in quasilinear time.
Abstract: S-machines were first introduced in a 1997 paper (published only in 2002) by Sapir, Birget, and Rips, where they were used to prove that a finitely generated group \(G\) with a word problem of complexity \(T(n)\) can be embedded into a finitely presented group \(H\) with Dehn function of \(G\) in \(H\) at most \(T(n)^4\). This bound hinges on the fact that S-machine can emulate a Turing machine in time \(T(n)^3\). We improve this emulation bound to \(T(n)^{1+\varepsilon}\) for any \(\varepsilon > 0\), which, hopefully, implies that the bound on the Dehn function can be improved to \(T(n)^{2+\varepsilon}\).
George Domat (University of Utah)
Title: A Graph Analogue for Big Mapping Class Groups and Coarse Geometry
Abstrct: We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina which hopes to answer the question: What is "Big Out(F_n)"? This group will consist of proper homotopy classes of proper homotopy equivalences of locally finite, infinite graphs. We will then discuss work in progress with Hannah Hoganson and Sanghoon Kwak on attempts to classify the coarse geometry of these groups.
Michael Hull (University of North Carolina, Greensboro)
Title: Generalized graph manifolds and the Singer Conjecture
Abstract: We prove the Singer Conjecture for extended graph manifolds (in the sense of Frigerio-Lafont-Sisto) and pure complex-hyperbolic higher graph manifolds (in the sense of Connell-Suarez-Serrato) with residually finite fundamental group. The proof is based on Price inequalities developed by Di Cerbo-Stern. This is joint work with L. Di Cerbo