In these days, the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.
— H. Weyl, Invariants; Duke Math. J., 5 (1939), 489–502.
In these days, the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.
— H. Weyl, Invariants; Duke Math. J., 5 (1939), 489–502.
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.
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: TBA
Abstract: TBA
Host: Dan Margalit
Wednesday, November 5
Daniel Groves (University of Illinois, Chicago)
Title: Drilling in hyperbolic 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 also discuss work in preparation where we apply this idea to study surface subgroups, and speculate about further applications. This is joint work with P. Haissinky, J. Manning, D. Osajda, A. Sisto, and G. Walsh.
Host: Talia Fernos
Wednesday, November 12
Kevin Pilgrim (Indiana University)
Title: TBA
Abstract: TBA
Host: Dan Margalit
Wednesday, November 19
Tina Torkaman (University of Chicago)
Title: TBA
Abstract: TBA
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
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, January 21
TBA
Wednesday, January 28
TBA
Wednesday, February 4
TBA
Wednesday, February 11
TBA
Wednesday, February 18
Anush Tserunyan (McGill University)
Title: TBA
Abstract: TBA
Host: Denis Osin
Wednesday, February 25
TBA
Wednesday, March 4
TBA
Wednesday, March 18
TBA
Wednesday, March 25
TBA
Wednesday, April 1
TBA
Wednesday, April 8
TBA
Wednesday, April 15
TBA