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 2024 - Spring 2025 

Wednesday, August 28

Filippo Calderoni (Rutgers)

Title: Descriptive set theory and the L-space conjecture 

Abstract: In this talk, we analyze the Borel complexity of the spaces of left-orders for left-orderable groups modulo conjugacy. In particular, we will focus on fundamental groups of three-manifolds.  Our results show that the space of left-orders for fundamental groups of knots is positively complicated. Moreover, we will see how the Borel complexity for the fundamental groups of a large class of three-manifolds is tightly related to the L-space conjecture. This is joint work with Adam Clay. 

Host: Denis Osin

Wednesday, September 4

Katherine Booth (Vanderbilt)

Title: Automorphisms of the smooth fine curve graph 

Abstract: The smooth fine curve graph of a surface is an analogue of the fine curve graph that only contains smooth curves. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves. 

Wednesday, September 11

Noah Caplinger (University of Chicago)

Title: Solvable Baumslag-Solitar Lattices 

Abstract: The solvable Baumslag-Solitar groups BS(1,n) each admit a canonical geometric model space, Xn. I will talk about the group of isometries Isom+(Xn), with particular focus on (the rigidity of) its lattices. 

Host: Dan Margalit

Wednesday, September 18

Christian Rosendal (University of Maryland)

Title: Asymptotically spherical groups 

Abstract: The fundamental observation of geometric group theory is of course that every finitely generated discrete group carries an inherent quasimetric structure, that is, a metric that is canonically defined up to quasi-isometry. As has become clear over the last decade, the same happens in a much wider setting, including for much larger topological groups with small generating sets. In response to some speculations by Sebastian Hurtado, we show that for a large class of discrete and topological groups there is a more refined inherent geometry, which provides an asymptotic notion of spheres. We will provide examples of such groups and also deduce some of the consequences of this refined structure. This is joint work with Jenna Zomback. 

Host: Denis Osin

Wednesday, September 25

Brandis Whitfield (Temple University)

Title: Short curves of end-periodic mapping tori 

Abstract: A homeomorphism of a infinite-type surface is end-periodic if each of its ends is either attracting or repelling under the map. Surprisingly, the end-periodicity of the map implies that its associated mapping torus is tame, i.e. the interior of a compact manifold. Further, if the map is atoroidal, then its mapping torus admits a hyperbolic structure. As an "infinite type" analogue to work of Minsky in the finite-type setting, we show that given a subsurface Y of S, the subsurface projections between the "positive" and "negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of Y as it resides in Mf. In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic 3-manifolds, show these techniques may be used in the infinite-type setting, and how our main theorems give back results to the closed, fibered setting. 

Host: Spencer Dowdall

Wednesday, October 2

Dickmann Ryan (Vanderbilt)

Title: Big mapping class groups of surfaces and graphs

Abstract: We will discuss the mapping class groups of infinite-type surfaces, known as big mapping class groups, and how they relate to the mapping class groups of infinite locally finite graphs as introduced by Algom-Kfir and Bestvina. This is joint work with Hannah Hoganson and Sanghoon Kwak. 

Monday, October 7

Xiao Chen (Tsinghua University)

Title: Maximal Systems of Curves on Surfaces

Abstract: Curve complexes and arc complexes are used to understand mapping class groups and Teichmüller spaces. Simplices of curve complexes and arc complexes are labeled by systems of simple loops and systems of simple arcs on surfaces. In particular, the cardinalities of maximal systems of curves determine the dimension of these complexes. In this talk, we explain how to determine the cardinalities of maximal (complete) 1-systems of curves on a non-orientable surface.

Host: Dan Margalit

Wednesday, October 9

Koichi Oyakawa (Vanderbilt)

Title: Geometry and dynamics of the extension graph of graph product of groups

Abstract: In this talk, I introduce the extension graph of a graph product of groups and explain its geometry. This notion enables us to study the properties of graph products by exploiting the large-scale geometry of its defining graph. In particular, I show that the asymptotic dimension of the extension graph exhibits the same behavior as in the case of quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. In addition, I present applications of the extension graph to the study of convergence actions, graph wreath products, and group von Neumann algebras when the defining graph is hyperbolic. 

Wednesday, October 16

Sierra Knavel (Georgia Tech)

Title: All about genus-2 Lefschetz fibrations

Abstract: Donaldson proved that symplectic 4-manifolds have the structure of a Lefschetz pencil which can be blown up to a fibration. Gompf later showed that any Lefschetz fibration has a symplectic structure. This one-to-one correspondence signifies that understanding Lefschetz fibrations is a good way to understand symplectic 4-manifolds. In this talk, I will define genus-g Lefschetz fibrations and note their special properties when the genus of its fiber is 2. Then, I will mention results concerning the fundamental groups of these objects and some progress towards classifying the non-holomorphic genus-2 Lefschetz fibrations. 

Host: Dan Margalit

Wednesday, October 23

Frank Wagner (Ohio State University)

Title: Malnormal subgroups of finitely presented groups

Abstract: We introduce a generalization of the computational model of S-machines and explore how it is employed to obtain the following refinement of Higman's embedding theorem: Every finitely generated recursively presented group may be embedded as a malnormal subgroup of a finitely presented group.  The properties of this embedding are then examined; for example, that the embedding is a quasi-isometry, producing a refinement of several theorems of Olshanskii.  Finally, we discuss how the embedding may be constructed to preserve the decidability of the word problem, yielding a refinement of a theorem of Clapham.

Host: Denis Osin

Wednesday, October 30

Anna Marie Bohmann (Vanderbilt) 

Title: Scissors congruence and K-theory of covers 

Abstract: Scissors congruence is an equivalence relation on polytopes that we get by declaring two polytopes to be equivalent when we can one up and reassemble to get the other.  The question of when two polytopes are equivalent under this relation is the subject of Hilbert's third problem, first addressed by Dehn in 1901.  Later in the 20th century, Dupont and Sah approached the question of scissors congruence by building "scissors congruence groups" that encode this relation. New approaches to algebraic K-theory by Zakharevich allow us to define "higher scissors congruence groups" that encode not just whether polytopes are equivalent, but whether they are equivalent in different ways. I will discuss a new definition of higher scissors congruence groups that gives a natural map to group homology with which we can probe these groups. No knowledge of algebraic K-theory is expected.  This work is joint with Gerhardt, Malkiewich, Merling, and Zakharevich. 

Wednesday, November 6

Justin Malestein (University of Oklahoma) 

Title: The density of Penner-Thurston pseudo-Anosov mapping classes 

Abstract: A theorem of Nielsen-Thurston tells us that every mapping class of a closed surface (i.e. a homotopy class of homeomorphisms) is one of three types: periodic, reducible or pseudo-Anosov. Pseudo-Anosov mapping classes are precisely those with a representative preserving a pair of transverse (measured) foliations and they have been shown to be generic in the mapping class group due mainly to work of Maher and Rivin. One of the main ways to construct explicit pseudo-Anosov mapping classes is via the Penner-Thurston construction, and fairly recently, it was shown by Shin-Strenner and Verberne that not all pseudo-Anosov mapping classes arise from this construction. In this talk, I will discuss how dense/generic the Penner-Thurston pseudo-Anosovs are in some subsemigroups of the mapping class group. Joint work with Josh Pankau and Jing Tao. 

Host: Spencer Dowdall

Wednesday, November 13

Ruth Charney (Brandeis University)

Title:  Parabolic Subgroups of Artin Groups 

Abstract:  Artin groups have played a central role in geometric group theory and low-dimensional topology, yet many open questions remain about these groups.  In many cases, addressing these questions reduces to understanding properties of their parabolic subgroups.  In this talk, I will discuss the role of parabolic subgroups and several conjectures about these groups. By establishing certain relations between these conjectures, we obtain new examples of acylindrically hyperbolic Artin groups.  This is joint work with Alexandre Martin and Rose Morris-Wright.

Host: Denis Osin

Wednesday, November 20

Sahana Balasubramanya (Lafayette College)

Title: Non-recognizing spaces for stable subgroups

Abstract: We say an action of a group on a space recognizes all stable subgroups if every stable subgroup of G is quasi-isometrically embedded in the action on . The problem of constructing or identifying such spaces has been extensively studied for many groups, including mapping class groups and right angled Artin groups- these are well known examples of acylindrically hyperbolic groups. In these cases, the recognizing spaces are the largest acylindrical actions for the group. One can therefore ask the question if a largest acylindrical action of an acylindrically hyperbolic group (if it exists) is a recognizing space for stable subgroups in general. We answer this question in the negative by producing an example of a relatively hyperbolic group whose largest acylindrical action fails to recognize all stable subgroups. This is joint work with Marissa Chesser, Alice Kerr, Johanna Mangahas, and Marie Trin.

Host: Denis Osin

Wednesday, December 4

Giulio Tiozzo (University of Toronto)

Title: A characterization of hyperbolic groups via contracting elements

Abstract: The notion of a contracting element has become central in geometric group theory, singling out, in an arbitrary metric space, the geodesics that behave like geodesics in a delta-hyperbolic space. In this work, joint with K. Chawla and I. Choi, we prove the following characterization of hyperbolic groups: a group is hyperbolic if and only if the D-contracting elements are generic with respect to counting in the Cayley graph. 

Host: Denis Osin

Wednesday, January 22

Mike Mihalik  (Vanderbilt)

Title: Stallings’ group is simply connected at infinity

Abstract: For n ≥ 2, let Bn denote the kernel of the homomorphism from the direct product of n-copies of the free group on two generators to the group of integers which sends all generators to the generator 1. The groups Bn are called the Bieri-Stallings groups and Bn is of type Fn-1 but not Fn. Classical results can be used to show that Bn is (n-3)-connected at infinity for n ≥ 3. Since Bn is not type Fn it is not (n-1)-connected at infinity. Stallings’ proved that B2 is finitely generated but not finitely presented. We conjecture that for n ≥ 2, Bn is (n-2)-connected at infinity. For n = 2, this would mean that B2 is 1-ended and for n = 3 that B3 (typically called Stallings’ group) is simply connected at infinity. We verify the conjecture for n = 2 and n = 3. Our main result is the case n = 3: Stalling’s group is simply connected at infinity.

Wednesday, January 29

Abdul Zalloum (University of Toronto)

Title: From coarse to fine: a generalized Sageev's construction in hyperbolic spaces

Abstract: I will discuss a construction that starts with a set S, a collection of bi-partitions on S called walls, and produces a spectrum of metric spaces, including Helly graphs, hyperbolic spaces, and injective metric spaces. The resulting space depends on the combinatorics of the walls; for example, if pairs of points are separated by finitely many walls, the resulting metric space is exactly Sageev's dual CAT(0) cube complex. I will also discuss a range of applications of this construction, with focus on the most recent: in joint work with Petyt-Spriano, we use the above framework to show that residually finite hyperbolic groups admit globally stable cylinders, addressing a question of Rips-Sela who asked in 1995 whether torsion-free hyperbolic groups possess such cylinders.

Host: Dan Margalit

Wednesday, February 5

Itamar Vigdorovich (UCSD)

Title: Effective mixed identity freeness for higher rank lattices and applications to C*-algebras 

Abstract: An identity on a group G is a word w that holds throughout the entire group. If w is allowed to include coefficients from G (not just variables), it is called a mixed identity. We show that a lattice Γ in PSL(n, R) has no non-trivial mixed identities in a quantitative and uniform manner: for any r, there exists an element γ of linear length in r that violates all non-trivial mixed words of length at most r. This has powerful C*-algebraic applications due to the recent breakthrough of Amrutam, Gao, Kunnawalkam-Elayavalli, and Patchell. Indeed, when combined with the rapid decay property (which is known, for example, for all cocompact lattices in SL(3, R)), we deduce that the reduced C*-algebra of the lattice satisfies strict comparison, stable rank 1, K0-stability under ultrapowers, uniqueness of the Jiang-Su embedding, and other key properties essential for C*-classification purposes.

Host: Denis Osin

Wednesday, February 12

Jesse Peterson (Vanderbilt)

Title: Property HaHaHaHaHaHaHaHaHaHaHa... HaHaHaHaHaHaHaHaHa... HaHaHaHaHaHa... HaHaHa... 

Abstract: A seminal result of Haagerup from 1979 is that the word length function with respect to a free generating set on a free group is conditionally negative definite. Groups possessing a proper conditionally negative definite function have since been said to have the Haagerup property and comprise a large and well-studied class of groups containing all amenable groups in addition to free groups. The Haagerup property is a particularly nice analytic approximation property since it passes to subgroups, is a local property, and is stable under measure-equivalence and W*-equivalence. One drawback however is that it is not in general closed under extensions. In this talk, I will discuss a weakening of the Haagerup property, which still has many nice analytic properties, but is more flexible when it comes to constructions such as group extensions. In fact, I will discuss a hierarchy of such properties indexed by the countable ordinals. This is joint work with Fabian Salinas. 

Wednesday, February 19

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, February 26

Matt Zaremsky (SUNY Albany)

Title: Aut(Fn) satisfies the Boone-Higman conjecture

Abstract:  The Boone-Higman conjecture (1973) predicts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. The "if" direction is true and easy, but the "only if" direction has been open for over 50 years. The conjecture is known to hold for various families of groups, perhaps most prominently the groups GL_n(Z) (due to work of Scott in 1984), and hyperbolic groups (due to work of Belk, Bleak, Matucci, and myself in 2023). In this talk I will discuss some recent work joint with Belk, Fournier-Facio, and Hyde establishing the conjecture for Aut(Fn), the group of automorphisms of the free group Fn. I will also highlight an interesting sufficient condition for satisfying the conjecture, which just amounts to finding an action of the group with certain properties, with no need to actually deal with simple groups. 

Host: Denis Osin

Wednesday, March 5

Madeline Brandt  (Vanderbilt)

Title: Tropicalizations of Shimura Varieties 

Abstract: I will speak about current work with Assaf, Bruce, Chan, and Vlad, in which we construct tropicalizations of Shimura varieties and use these tropicalizations to compute weight zero compactly supported homology of some Shimura varieties. Many important examples of classical moduli spaces are Shimura varieties, including Ag, the moduli space of Abelian varieties. I will begin by explaining tropicalizations and how they can be used to say something about the cohomology of a variety, and then I will discuss Shimura varieties and their compactifications, with running examples.  

Wednesday, March 19 

Oishee Banerjee (Florida State University)

Title: Topology of Algebraic Mapping Spaces

Abstract: The topology of spaces of continuous maps has been a topic of great interest for decades. Meanwhile, through the Weil conjectures, the topology (specifically, ℓ-adic cohomology) of certain moduli spaces of algebraic functions has had a profound impact on number theory. In this talk, I will outline a framework for studying the cohomology of the moduli space of algebraic maps Alg(X, Y) under certain conditions on X and Y. Our methods, in particular, confirm the geometric Manin conjecture, which predicts an asymptotic estimate for the number of algebraic maps over finite fields, in the special case where Y is a complete simplicial toric variety and X is a curve. No prior knowledge of algebraic geometry beyond that of a typical first-year graduate student will be assumed.

Host: Dan Margalit

Wednesday, March 26

Caglar Uyanik  (University of Wisconsin, Madison)

Title: Singularity of Cannon-Thurston maps

Abstract: Cannon and Thurston showed that a hyperbolic 3-manifold that fibers over the circle gives rise to a sphere-filling curve. The universal cover of the fiber surface is quasi-isometric to the hyperbolic plane, whose boundary is a circle, and the universal cover of the 3-manifold is 3-dimensional hyperbolic space, whose boundary is the 2-sphere. Cannon and Thurston showed that the inclusion map between the universal covers extends to a continuous map between their boundaries, whose image is onto. In particular, any measure on the circle pushes forward to a measure on the 2-sphere using this map. We compare several natural measures coming from this construction.

Host: Spencer Dowdall

Wednesday, April 2

Talia Fernos  (Vanderbilt)

Title: Lattice envelopes and groups acting AU-acylindrically on products of hyperbolic spaces 

Abstract: In this joint work with Balasubramanya, we explore the capacity for a group acting AU-acylindrically on a finite product of delta-hyperbolic spaces to satisfy three properties introduced by Bader, Furman, and Sauer. When satisfied, these properties restrict the potential ambient group in which it can be imbedded as a lattice. In this talk, we will also discuss the classification of actions on a delta-hyperbolic space, the associated trifurcation of elliptic actions, and the relationship to normal and commensurate subgroups. We will end the talk with an open question. 

Wednesday, April 9

John Ratcliffe (Vanderbilt)

Title:  The G-index of a spin closed hyperbolic 4-manifold M

Abstract: In this talk, we will show how to compute the G-index of a spin closed hyperbolic 4-manifold M for a group G of symmetries of a spin structure on M. As an example, we will compute the G-index for the group G of symmetries of the fully symmetric spin structure on the Davis closed hyperbolic 4-manifold M. Our talk will involve finite groups, infinite discrete groups, and Lie groups.  The talk is based on joint work with Steven Tschantz.

Wednesday, April 16

Robbie Lyman (Rutgers)

Title: Graphical models for groups

Abstract: In geometric group theory, we love groups and graphs. Every (abstract) group has (many) Cayley graphs, each one associated with a choice of a generating set. Recently I've been curious about topological groups, inspired by the budding theory around and community of people inspired by mapping class groups of infinite-type surfaces and by Christian Rosendal's breakthrough work on geometries for topological groups. I'm hoping to share out some of what I've learned about topological groups acting on graphs. Much of this comes from recent joint work with Beth Branman, George Domat and Hannah Hoganson.

Host: Talia Fernos