Organizers: Spencer Dowdall, Dan Margalit, Denis Osin
Time: Wednesdays, 4:10–5:00pm
Location: SC 1432
Mailing List: If you would like to be added to the seminar mailing list and receive regular announcements, please contact one of the organizers.
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Fall 2025 - Spring 2026
Wednesday, September 10
Dan Margalit (Vanderbilt)
Title: Algorithms for pseudo-Anosov maps
Abstract: In joint work with Strenner, Taylor, and Yurttas, we give an algorithm whose input is a product of Dehn twists and whose output is the Nielsen-Thurston type of the associated mapping class, along with the associated data: period, reducing curves, stretch factor, and foliations. Our main theorem explains why Mark Bell's program Flipper works. We aim to make the talk accessible to students learning about mapping class groups.
Wednesday, September 17
Denis Osin (Vanderbilt)
Title: Simple p-adic Lie groups with abelian Lie algebras
Abstract: For each prime p and each positive integer d, we construct the first examples of second countable, topologically simple, p-adic Lie groups of dimension d with abelian Lie algebras. This answers a question asked by P.-E. Caprace and N. Monod. In my talk, I will survey the necessary background material and explain why this question is of fundamental importance in the theory of p-adic Lie groups. Perhaps surprisingly, the proof of the main result makes use of small cancellation techniques in groups acting on hyperbolic spaces. Geometric ideas come into play through a general construction that associates a non-discrete, totally disconnected topological group with any discrete group satisfying a certain algebraic condition. The talk is based on joint work with P.-E. Caprace and A. Minasyan.
Wednesday, September 24
Shuxian Song (Vanderbilt)
Title: Generating the Mapping Class Group
Abstract: Generating sets of mapping class groups have been a central theme in the study of surfaces and low dimensional topology. In this survey talk, I will begin with Dehn twist generators. We gave a short, self-contained proof that the Humphries generators indeed form a generating set for mapping class groups. Unlike other proofs, our argument did not introduce extraneous generators. I will then turn to generating sets consisting of torsion elements and pseudo-Anosov maps, surveying results on small generating sets by elements of a certain order or by pseudo-Anosov maps.
Wednesday, October 1
Nic Brody (UC Santa Cruz)
Title: Presentations for Linear Groups
Abstract: Suppose S is a finite set of invertible matrices over the field of algebraic numbers. Immediately, this offers a plethora of perspectives we can use to shed light on the group these matrices generate, and there is a rich interplay between these topics. In this talk we will survey three known methods for constructing subgroups of linear groups, and propose that there may not be any other possibilities! Furthermore, we will discuss the circumstances in which one can efficiently compute a presentation for a given finitely generated matrix group. We propose that there is always an algorithm for computing a presentation for a finitely generated subgroup of PGL_2 over the algebraic closure of Q. To study this, we devote special attention to the case in which the group is discrete in PSL_2(C); that is, the world of Kleinian groups and 3-manifolds.
Host: Talia Fernos
Wednesday, October 15
Yash Lodha (Purdue University)
Title: Two new constructions of finitely presented infinite simple groups
Abstract: I will describe two new constructions of finitely presented infinite simple groups. First, I will present a construction of finitely presented (and type F∞) simple groups that act by orientation preserving homeomorphisms on the real line. These are the first such examples. Next, I will present a construction of a family of finitely presented infinite uniformly simple groups, where the Ulam width can get arbitrarily large. Among the class of finitely generated (but not finitely presentable) groups, the existence of such examples was demonstrated in the work of Muranov from 2007. Our construction provides the first such family of examples in the class of finitely presented infinite groups. This is joint work with James Hyde.
Host: Denis Osin
Wednesday, October 22
Thomas Hill (University of Utah)
Title: Mapping class groups of infinite graphs
Abstract: The group of topological symmetries of a surface, known as the mapping class group, provides a powerful lens for studying the geometry and topology of surfaces. The outer automorphism groups of a free group, Out(Fn), can be thought of as a graph-theoretic analog of the mapping class group. Although results do not translate directly between the surface and graph settings, their structural parallels inspire questions and techniques between these areas. Over the past decade, there has been growing interest in studying big mapping class groups, namely those of infinite type surfaces. In this talk, I will discuss recent progress on the analog of Out(Fn) for infinite graphs, including an Ivanov-type rigidity theorem and results about the algebraic properties of their asymptotically rigid subgroups.
Host: Spencer Dowdall
Wednesday, October 29
Annie Holden (Notre Dame)
Title: Cohomology of Handlebody Torelli Groups
Abstract: We begin by introducing the Torelli subgroup of the mapping class group of a surface and outlining known results and its low-dimensional cohomology. We then present recent work extending these results to two Torelli subgroups of the mapping class group of a handlebody.
Host: Dan Margalit
Wednesday, November 5
Talia Fernos (Vanderbilt)
Title: A plethora of examples, AU-acylindricity in higher rank
Abstract: This talk will be an overview of the class of groups (and associated subdirect products) that admit acylindrical actions of Ambiguous Uniformity on finite products of delta-hyperbolic spaces. The primary focus will be the semi-simple case, i.e. when the factor actions are of general type. While based on joint work with Balasubramanya, this talk is a survey.
Wednesday, November 12
Spencer Dowdall (Vanderbilt)
Title: Complexity length and lattice point counting in Teichmüller space
Abstract: This talk expands on the mini-colloquium I gave on September 25. I will discuss the key geometric features of Teichmüller space that are relevant for counting problems. These include thin regions and their inherent product structure, curve complex subsurface projections, and Rafi's distance formula. I will then introduce a new notion of "complexity length" in Teichmüller space that carefully accounts for how geodesics move through product regions and lends itself to counting problems. Using this tool, we are able to count the number of mapping class group lattice points for various types of types of elements. Joint work with Howard Masur.
Wednesday, November 19
Tina Torkaman (University of Chicago)
Title: Random Measured Laminations and Teichmüller Space
Abstract: In this talk, we introduce a canonical geodesic current KX for each X∈Tg, representing a randomly chosen simple closed geodesic on X. We establish results analogous to Bonahon's work on the Liouville measure. In particular, we show that the map X↦KX defines a proper embedding of Teichmüller space Tg into the space of geodesic currents. This embedding leads to a compactification of Tg that differs from Thurston's compactification (which Bonahon's results yield via the Liouville measure). We will discuss the construction of KX and its geometric properties. This is joint work with Curt McMullen.
Host: Spencer Dowdall
Wednesday, December 3
Darren Creutz (Vanderbilt)
Title: Harmonic Functions on Groups
Abstract: Implicit in the work of Mok (1995) and Kleiner (2010) is the statement that a finitely generated group is infinite if and only if there exists a nonconstant harmonic (with respect to the counting measure on a generating set) function on the group. I will present (sort of) recent work generalizing this to locally compact groups under a more general class of measures: such a group is noncompact if and only if it admits a nonconstant Lipschitz-bounded harmonic function (for any/every reasonable probability measure). The proof involves an improvement of Mok/Kleiner's energy argument. I will also present how this improvement makes progress towards the Margulis-Zimmer conjecture on commensurated subgroups of lattices in higher-rank semisimple groups.
Wednesday, January 14
Sarah Maloni (University of Virginia)
Title: Geometric Structures in Higher Teichmüller Theory
Abstract: The Teichmüller space of a surface S is the deformation space of marked hyperbolic structures. This can be viewed as a component of representations of the fundamental group π1(S) into the isometry group of hyperbolic space. Higher Teichmüller Theory generalizes this idea by studying representations of surface groups into Lie groups of higher rank. In this talk, we will introduce key ideas in the theory of deformations of geometric structures, (higher) Teichmüller Theory, and the concept of Anosov representations. We'll also discuss recent work with Alessandrini, Tholozan, and Wienhard and work of my PhD student Mason Hart, showing how these representations also relate to deformations of geometric structures.
Host: Spencer Dowdall
Wednesday, January 21
Lvzhou Chen (Purdue University)
Title: The Wiegold problem and the weight of free products
Abstract: The weight (or normal rank) of a group G is the smallest number of elements that normally generate G. This notion plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking at the abelianization. The Wiegold problem asks if there is a finitely generated perfect group that has weight greater than one, namely the group cannot be normally generated by any single element. In a joint work with Yash Lodha, we show that free products of nontrivial left-orderable groups all have weight greater than one, which solves the Wiegold problem. I will explain the topological and dynamical ingredients in the proof of our theorem.
Host: Denis Osin
Wednesday, January 28
Cancelled due to inclement weather
Wednesday, February 4
Koichi Oyakawa (McGill)
Title: Hyperfiniteness of the boundary action of virtually special groups
Abstract: A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure the complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection between Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the induced orbit equivalence relation on the Roller boundary is hyperfinite.
Host: Denis Osin
Wednesday, February 11
Juhun Baik (Vanderbilt)
Title: Topological normal generation of big mapping class groups
Abstract: Finding a normal subgroup/generator of mapping class groups has been studied by many experts (H. Baik, Clay, D. Kim, S.Kim, Koberda, Lanier, Mangahas, Margalit, Wu, ...). In the case of closed surfaces, if the genus of a given surface is more than 2, by Lanier-Margalit, any pseudo-Anosov map of stretch factor less than \sqrt{2} or a finite-order element except hyperelliptic involutions normally generates the mapping class group. We explore the same question regarding big mapping class groups, the mapping class groups of infinite-type surfaces. In this talk, I will first introduce the topology of big mapping class groups and highlight differences from the finite type cases. I will then identify when the big mapping class group is topologically normally generated, and give an upper bound on how many generators are needed to topologically normally generate the whole group.
Wednesday, February 18
Anush Tserunyan (McGill University)
Title: TBA
Abstract: TBA
Host: Denis Osin
Friday, February 20 (2:30-3:30, SC1313)
Kevin Pilgrim (Indiana University)
Title: Hurwitz bisets: dynamics and topology
Abstract: Given a finite subset $P \subset S^2$, consider the set $\text{BrMod}(S^2,P)$ of isotopy classes of orientation-preserving, branched self-covers $f: (S^2, P) \to (S^2, P)$ whose branch values lie in $P$ and such that $f(P) \subset P$. The (pure) mapping class group $\text{PMod}(S^2,P)$ acts naturally by pre- and post-composition on $\text{BrMod}(S^2,P)$. A two-sided orbit we call a \emph{Hurwitz biset}. Hurwitz bisets are rich in algebraic structure; they act naturally on Teichmüller spaces; and they are closely connected with the combinatorics and dynamics of the representing maps they contain. I will give a gentle overview, concluding with some open non-dynamical problems that are in turn connected to topics like unlikely intersections in algebraic geometry and special configurations of torsion points on elliptic curves.
Host: Dan Margalit
Wednesday, February 25
TBA
Wednesday, March 4
Daniel Groves (UIC)
Title: Drilling in hyperbolic groups, and surface subgroups in hyperbolic PD(3) groups
Abstract: We define the notion of "drilling" a hyperbolic group to find a relatively hyperbolic group, and discuss applications to the Cannon Conjecture. I will discuss work in preparation where we apply this idea to study surface subgroups of residually finite hyperbolic groups with two-sphere boundary. This is joint work with P. Haissinsky, J. Manning, D. Osajda, A. Sisto, and G. Walsh, and the work in preparation is joint with G. Walsh.
Host: Talia Fernos
Wednesday, March 18
Junhwi Lim (Vanderbilt)
Title: Interesting von Neumann algebras coming from property (T) groups
Abstract: Groups give rise to rich examples of von Neumann algebras, and conversely, von Neumann algebras provide a natural framework for studying group representations. Analogously to subgroup index, the Jones index measures how large a von Neumann algebra is relative to its subalgebra. However, the set of all possible Jones indices is known only for a few examples. In particular, de la Harpe proposed a problem of identifying the Jones indices for von Neumann algebras associated with property (T) groups. We show that property (T) wreath-like product groups introduced by Chifan-Ioana-Osin-Sun give rise to von Neumann algebras whose Jones index sets consist of all positive integers. No background on von Neumann algebras is assumed. This talk is based on joint work with Ionut Chifan.
Wednesday, March 25
Damian Osajda (University of Wroclaw)
Title: TBA
Abstract: TBA
Host: Denis Osin
Wednesday, April 1
TBA
Wednesday, April 8
Beeri Greenfeld (Hunter College, CUNY)
Title: TBA
Abstract: TBA
Host: Denis Osin
Wednesday, April 15
TBA
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