2025-10-16 at 13:30:00 in B3.02
Speaker: Davide Spriano (University of Warwick)
Title: Curtains, walls and stable cylinders.
Abstract: In this talk we will discuss a generalization of Sageev’s wallspace construction that allows to study the geometry of certain spaces by combinatorial properties of certain walls. Specifically, we’ll look at the interactions with hyperbolicity and focus on two applications. In CAT(0) spaces, these techniques allow to construct a “universal hyperbolic quotient”, called the curtain model, that is analogous to the curve graph of a surface. When focusing on a space that is already hyperbolic, the construction can be used to improve its fine properties, and in particular we address a conjecture of Rips and Sela and show that residually finite hyperbolic groups admit globally stable cylinders. This is joint work with Petyt and Zalloum.
2025-10-23 at 13:30:00 in B3.02
Speaker: Mahan Mj (Tata Institute of Fundamental Research)
Title: Hyperbolic and elliptic commensurations
Abstract: A group G is said to commensurate a subgroup H, if for all g in G, H^g \cap H is of finite index in H and H^g, where H^g denotes the conjugate of H by g. The commensuration action of G on H can be studied dynamically. This gives rise to two extreme behaviors: hyperbolic and elliptic. We will discuss what these mean and survey a range of theorems and conjectures in this context, starting with work of Mostow and Margulis, and coming to the present day.
2025-11-06 at 13:45:00 in b3.02
Speaker: Islam Foniqi (University of East Anglia)
Title: Submonoid Membership Problems in One-Relator Groups and Monoids, Surface Groups, and Beyond.
Abstract: The word problem for one-relator monoids remains one of the long-standing open questions in combinatorial algebra. One way to approach it is by studying related decision problems, in particular the submonoid membership problem, in both monoids and groups. In this talk, I will discuss how these problems are connected, drawing on classical work by Adian and Guba. I will also highlight the role of embeddings of right-angled Artin groups and trace monoids, which offer useful insights into the structure of these questions. Finally, I will present recent joint results with Robert D. Gray on the submonoid membership problem in surface groups, and in the broader hyperbolic setting.
2025-11-13 at 13:30:00 in B3.02
Speaker: Will Cohen (University of Cambridge)
Title: Improving acylindrical actions on trees
Abstract: Loosely speaking, an action of a group on a tree is acylindrical if long enough paths must have small stabilisers. Groups admitting such actions form a natural subclass of acylindrically hyperbolic groups, and an interesting feature of acylindrical actions on trees is that many properties of groups are inherited from their vertex stabilisers. In order to make use of this, it is important to have some degree of control over these stabilisers. For example, can we ask for these stabilisers to be finitely generated? Even stronger, if our group is hyperbolic, can we ask for the stabilisers to be quasiconvex?In this talk, I will introduce acylindrical actions and some stronger and related concepts, and discuss a method known as the Dunwoody—Sageev resolution that we can use to move between these concepts and provide positive answers to the above questions in some cases.
2025-11-20 at 13:30:00 in B3.02
Speaker: Sebastian Hensel (LMU Munich)
Title: Dynamics of torus homeomorphisms and the fine curve graph
Abstract: The fine curve graph is a Gromov hyperbolic graph on which the homeomorphism group of a surface acts. In this talk we will discuss joint work with Frédéric Le Roux which relates the surface dynamics of a torus homeomorphism to its action on the fine curve graph. We show in particular that the shape of a “big" rotation set is determined by the fixed points on the Gromov boundary of the graph. A key ingredient is a metric version of the WPD property for the homeomorphism group of the torus.
2025-11-27 at 13:30:00 in B3.02
Speaker: Jing Tao (University of Oklahoma)
Title: Tame maps of surfaces of infinite type
Abstract: A cornerstone in low-dimensional topology is the Nielsen-Thurston Classification Theorem, which provides a blueprint for understanding homeomorphisms of compact surfaces up to homotopy. However, extending this theory to non-compact surfaces of infinite type remains an elusive goal. The complexity arises from the behavior of curves on surfaces with infinite type, which can become increasingly intricate with each iteration of a homeomorphism. To address some of the challenges, we introduce the notion of tame maps, a class of homeomorphisms that exhibit non-mixing dynamics. In this talk, I will present some recent progress on extending the classification theory to such maps. This is joint work with Mladen Bestvina and Federica Fanoni.
2025-12-04 at 13:30:00 in B3.02
Speaker: Jannis Weis (KIT)
Title: From finiteness properties to polynomial filling via homological algebra
Abstract: If a group has type FP_n one can define higher filling functions, which give a quantitative refinement of FP_n by measuring the size of fillings of k‑cycles (k ≤ n). We develop a homological‑algebra framework that extends existing tools for finiteness properties to produce polynomial bounds for these filling functions. The goal is to make deducing polynomiality as straightforward as proving FP_n. This is based on joint work with Roman Sauer.
2025-12-11 at 13:30:00 in B3.02`
Speaker: Philipp Bader (University of Glasgow)
Title: Teichmüller curves via the Hurwitz-Hecke construction
Abstract: Teichmüller curves are totally geodesic algebraic curves inside the moduli space of Riemann surfaces of genus g. There are fascinating connections between Teichmüller curves and billiard flows on polygons.Given a Teichmüller curve, there is a way to construct another one in higher genus by taking a branched cover. If a Teichmüller curve does not arise in this way, we call it primitive. The classification of primitive Teichmüller curves is a problem that has been widely explored in the past decades but still leaves many questions unanswered. In fact, only in genus 2 there exists a complete classification. In every genus starting from 5 and higher only finitely many examples of primitive Teichmüller curves have been found.In this talk, we introduce the notions described above and present the so-called Hurwitz-Hecke construction; a method that can be used to construct Teichmüller curves. We will see that this construction gives rise to many of the known examples of Teichmüller curves. This is joint work in progress with Paul Apisa and Luke Jeffreys.
2026-01-15 at 13:30:00 in B3.02
Speaker: Jerónimo García Mejía (University of Warwick)
Title: Complete classification of the Dehn functions of Bestvina—Brady groups
Abstract: Introduced by Bestvina and Brady in 1997, Bestvina–Brady groups form a rich class of examples in geometric group theory, notable for their exotic finiteness properties. In this talk, I will focus on the Dehn functions of finitely presented Bestvina—Brady groups. The Dehn function of a group is a fundamental quasi-isometry invariant that provides a quantitative measure of the group's finite presentability. After recalling previous results, I will present a classification of the Dehn functions of Bestvina—Brady groups. This talk is based on joint work with Yu-Chan Chang and Matteo Migliorini.
2026-01-22 at 13:30:00 in B3.02
Speaker: Nansen Petrosyan (University of Southampton)
Title: The kernel knows
Abstract: For a graph product of groups, the canonical map to the direct product of the vertex groups has a kernel whose structure is not immediately apparent. Remarkably, this kernel turns out to be oblivious to most of the algebra one builds into the construction, yet it is sensitive to the underlying combinatorics.This has applications to the Baum--Connes conjecture, Brown's question, the Eilenberg--Ganea conjecture and inheritance properties of graph products of groups.This is joint work with Ian Leary.
2026-01-29 at 13:30:00 in B3.02
Speaker: Carolyn Abbott (Brandeis University)
Title: A non-acylindrically hyperbolic Morse local-to-global group
Abstract: Hyperbolic spaces satisfy two defining properties: every geodesic is Morse, and every local quasi-geodesic (on a sufficiently large local scale) is a global quasi-geodesic. In non-hyperbolic spaces, some geodesics may be Morse and others not, and local quasi- geodesics may or may not be global quasi-geodesics. Intuitively, the Morse geodesics pick out the “hyperbolic-like” directions in the space. Morse local-to-global (MLTG) spaces generalize hyperbolic spaces by requiring that the local-to-global property for hyperbolic spaces holds for all Morse elements. MLTG groups (groups whose Cayley graph is a MLTG space) include hyperbolic groups, Zn, many classes of Artin groups, including right-angled, mapping class groups, and groups hyperbolic relative to MLTG groups. All known examples that contain a Morse element are acylindrically hyperbolic; in light of which Russell, Spriano, and Tran asked whether this was always the case. In this talk, I’ll explain why the answer to this question is no by describing the construction of a non-acylindrically hyperbolic MLTG group with a Morse element. Along the way, I’ll describe how to generalize several techniques from small cancellation groups to general finitely generated groups. This is joint work with Stefanie Zbinden.
2026-02-05 at 13:30:00 in B3.02
Speaker: Luke Jeffreys (University of Bristol)
Title: Euler characteristics of SL(2,Z)-orbit graphs of square-tiled surfaces
Abstract: Square-tiled surfaces (aptly named) are surfaces obtained by gluing together a collection of unit squares along their sides (the square torus being the simplest example). These surfaces are special cases of translation surfaces, whose moduli space carries a natural action of the group SL(2,R). The famous works of Eskin–Mirzakhani and Eskin–Mirzakhani–Mohammadi were concerned with understanding the orbits of this action.This SL(2,R)-action restricts to an action of SL(2,Z) on square-tiled surfaces, and the orbits under this restricted action can be transformed into finite regular graphs. It is a long-standing conjecture of McMullen that a specific family of these orbit graphs in genus two forms a family of expander graphs. Providing indirect evidence for this conjecture, I will describe joint work with Carlos Matheus in which we prove that the absolute values of the Euler characteristics of the graphs in this family go to infinity (a requirement for any expander family). To do this, we are required to count a variety of objects, including integer points on algebraic hypersurfaces, pseudo-Anosov homeomorphisms of fixed dilatation, and orbifold points on certain algebraic curves in moduli space.
2026-02-06 at 11:00:00 in B2.04
Speaker: Genevieve Walsh (Tufts University)
Title: Drilling and what it is good for.
Abstract: A drilling of a group is a relatively hyperbolic group pair, which is a generalization of the fundamental group of a drilled manifold. We define this and show that drillings exist for residually finite hyperbolic groups with two-sphere boundaries. We also present some preliminary work which shows drilling can be used to understand the non-quasi convex surface subgroups of a residually finite hyperbolic group with 2-sphere boundary. The first part ofthis talk is joint work with D. Groves, P. Haïssinsky, J. Manning, D. Osajda, and A. Sisto. The second part is joint with D. Groves.
2026-02-12 at 13:30:00 in B3.02
Speaker: Giovanni Italiano (University of Oxford)
Title: TBA
2026-02-19 at 13:30:00 in B3.02
Speaker: Gabriel Corrigan (University of Glasgow)
Title: Finiteness properties of some automorphism groups of right-angled Artin groups
Abstract: Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that in fact it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.
2026-02-25 at 15:00:00 in B1.01
Speaker: Thomas Ng (Brandeis University)
Title: Hyperbolic quotients of hyperbolic spaces
Abstract: Hyperbolicity in groups was introduced by Gromov in the 80s as a simultaneous generalization of fundamental groups of negatively curved Riemannian manifolds and groups with efficient algorithmic complexity coming from small cancellation. Natural groups from low-dimensional topology including irreducible right-angled Artin and Coxeter groups, mapping class groups of compact surfaces, and outer automorhisms of free groups often fail to be hyperbolic, despite admitting meaningful actions nonpositively curved metric spaces. We will discuss geometric analog of a presentation 2-complex for studying the geometry of certain quotients of acylindrically hyperbolic groups. I will explain how these ideas are used to establish that appropriate models of random quotients preserve several notions of negative and non-positive curvature in groups.
2026-02-26 at 13:30:00 in B3.02
Speaker: Matteo Migliorini (Karlsruher Institut für Technologie)
Title: The Dehn function of Thompson's group T
Abstract: Thompson’s groups, introduced by Thompson in 1965, have had a lot of attention in the last fifty years. Being finitely presented, a natural question is to compute their Dehn function. All three groups are conjectured to have quadratic Dehn function; this conjecture was confirmed for Thompson’s group �� by Guba in 2006. During this talk, we show how to deduce from Guba’s result that Thompson’s group �� has quadratic Dehn function as well.
2026-03-05 at 13:30:00 in B3.02
Speaker: Tobias Hartnick (Karlsruher Institut für Technologie)
Title: TBA
2026-03-12 at 13:30:00 in B3.02
Speaker: Layne Hall (University of Warwick)
Title: Flows and ideal triangulations of three-manifolds
Abstract: Pseudo-Anosov flows are a class of dynamical systems on three-manifolds with deep connections to the topology and geometry of their underlying spaces. For instance, features of the flow are closely related to the manifold’s hyperbolic geometry, embedded surfaces, and fundamental group. A modern approach to studying these flows is with veering triangulations. These are rigid combinatorial objects that have provided new computational and algorithmic techniques to study the flows. In this talk, I will first give a broad overview of the correspondence between these flows and triangulations. I will then discuss my work on more flexible triangulations that capture a larger class of pseudo-Anosov flows and yield new explicit examples.
2026-05-07 at 13:30:00 in B3.02
Speaker: Edgar Bering (San Jose State University)
Title: Modulii of solenoids and commensurations of the free group
Abstract: The abstract commensurator of a group Comm(G) is the group of isomorphisms between finite-index subgroups, up to the equivalence of agreement on a common finite-index domain. Commensurators famously appear in Margulis' arithmeticity theorem, and for a suitable Lie group the abstract commensurator detects the arithmeticity of lattices. Notably, PSL_2(R) is not suitable: every lattice is virtually a free or closed surface group, and the abstract commensurator fails to detect arithmeticity in this setting. Other methods are required to study the abstract commensurator of finitely generated free or surface groups. Biswas, Nag, and Sullivan developed a ``universal Teichmuller theory" and approached the abstract commensurator of a surface group with ideas from mapping class group theory. I will introduce, following Culler and Vogtmann's outer space, the modulii space of metric graph solenoids. This space is equivalent to an appropriate direct limit of auter spaces (base points turn out important), and has a global structure coming from Leighton's theorem on graph covers. Once the audience is situated in this space I will use the natural action of the abstract commensurator Comm(F) on this space to reveal exciting features of Comm(F). This is joint work with Daniel Studenmund.
2026-05-14 at 13:30:00 in B3.02
Speaker: David Hume (University of Birmingham)
Title: The Poincaré critical exponent
Abstract: The Poincaré critical exponent (PCE) assigns to each bounded degree graph X a value p_X ∈ [1,+∞]. PCEs satisfy a strong monotonicity property: whenever there is a Lipschitz, bounded-to-one map X-> Y, it holds p_X ≤ p_Y. In particular, if X and Y are Cayley graphs of finitely generated groups H and G, and H is quasi-isometric to a subgroup of G, then p_X≤ p_Y.When X is the Cayley graph of a virtually nilpotent group G, p_X is the degree of polynomial growth, and for many hyperbolic groups p_X is the conformal dimension of the boundary of G.In this talk I will define PCEs, explain some of their properties and highlight many interesting open questions.
2026-05-15 at 13:30:00 in B3.02
Speaker: David Hume (CANCELLED)
Title: TBA
2026-05-21 at 13:30:00 in B3.02
Speaker: Jonathan Fruchter (University of Bonn)
Title: Virtual homological torsion in low dimensions
Abstract: A long-standing conjecture of Bergeron and Venkatesh predicts that in closed hyperbolic 3-manifolds, the amount of torsion in the first homology of finite-sheeted normal covers should grow exponentially with the degree of the cover as the covers become larger, at a rate reflecting the volume of the manifold. Yet no finitely presented residually finite group is known to exhibit such behaviour, and meaningful lower bounds on torsion growth are rare.In this talk I will explain how a particular two-dimensional lens offers a clearer view of some of the underlying mechanisms that create homological torsion in finite covers, and how they might relate to its growth. If time allows, I will also discuss how these ideas connect to the question of profinite rigidity: how much information about a group is encoded in its finite quotients.
2026-05-28 at 13:30:00 in B3.02
Speaker: Harsh Patil (University of Bristol)
Title: TBA
2026-06-04 at 13:30:00 in B3.02 Zeeman
Speaker: Vaibhav Gadre (Glasgow)
Title: Cannon--Thurston curves do not equidistribute
Abstract: Thurston showed that a fibered three-manifold is hyperbolic if and only if it is a mapping torus of a surface by a pseudo-Anosov map. For such a three-manifold, the inclusion of the fibre (at the level of the universal covers) gives an exponentially distorted copy of a hyperbolic plane into hyperbolic three-space. Nonetheless, Cannon--Thurston showed that one still obtains a continuous map from the circle at infinity (for the hyperbolic plane) to the sphere at infinity (for hyperbolic three-space), resulting in a space filling curve. This fits in a much broader context of Cannon--Thurston maps as for example studied by Mahan and others. We show that in contrast to classical Peano type curves, the Cannon--Thurston curves do not equidistribute: a large class of measures on the circle pushes forward to singular measures on the sphere.This is joint work with Maher, Pfaff, and Uyanik,
2026-06-11 at 13:30:00 in B3.02
Speaker: Abigail Hollingsworth (University of Warwick)
Title: Angle structures and representations through Dehn surgery space
Abstract: Let T be an ideal triangulation of a three-manifold M. The shapes of the tetrahedra in the triangulation give gluing equations, the solutions to which we call the Shape variety. We look at the real locus of the shape variety where the tetrahedra are all flat. There are three types of angle structure that exist with flat tetrahedra.After Thurston sketched the boundary of Dehn surgery space of the figure-eight knot, Hodgson broke it down into three types, which we denote as representations from the fundamental group of the knot complement into PSL(2,R), PGL(2,R) and SO(3).What different angle structures have which types of representation? Can they then also be deformed into a positive volume manifold such that all tetrahedra are positive? We will look at all the different combinations and determine which ones can exist.
2026-06-18 at 13:30:00 in B3.02
Speaker: Misha Schmalian (University of Oxford)
Title: TBA
