Geometry and Topology

Please contact Saul Schleimer, Davide Spriano, or Robert Kropholler if you would like to speak or to suggest a speaker. A picture of a cat that is made out of cubes

We will also attempt to maintain an up-to-date listing at researchseminars.org.

The seminar will be hybrid, and will be run weekly. The talk is in B3.02 Zeeman Buildingon Thursdays, times below. We will open and close the Zoom session on the hour. All of the talks will be streamed at this link.

Click on a title to view the abstract!

Upcoming Seminars

Past Seminars

11 December 2025 at 13:30 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.

04 December 2025 at 13:30 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.

27 November 2025 at 13:30 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.

20 November 2025 at 13:30 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.

13 November 2025 at 13:30 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.

06 November 2025 at 13:45 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.

23 October 2025 at 13:30 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.

16 October 2025 at 13:30 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.