Surface Dynamics and Fine Curve Graphs

Surface Dynamics and Fine Curve Graphs
May 26-29, 2026

Schedule

	Tuesday	Wednesday	Thursday	Friday
9:30-10:30	Richard Webb	Alejandro Kocsard	Alejo García-Sassi	Emmanuel Militon
11:00-12:00	Nelson Schuback	Katherine Booth	Nastaran Einabadi	Jing Tao

14:00-15:00	Problem session	Reading groups	Reading groups
15:30-16:30	Nicholas Vlamis	Reading groups	Reading groups

Coffee breaks will take place in Room 15-16-417. We will also have a reception there, on Tuesday at 18:00.

Location
Room 15-16-413 on Tuesday, 15-25-502 on Wednesday and Friday and 15-24-102 on Thursday
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Campus Pierre et Marie Curie (Jussieu), Sorbonne Université
4, place Jussieu, 75252 Paris

Titles and abstracts

Katherine Booth: Subgroups of the Homeomorphism Group and Smooth Fine Curve Graphs
Work of Ivanov connects the extended mapping class group of a surface to the automorphisms of the curve graph. For fine curve graphs, a much richer story is starting to evolve. While the homeomorphism group of a surface is isomorphic to the automorphisms of the fine curve graph with topological curves, restricting to smoother curves results in a previously unknown family of subgroups. In this talk, I will discuss smooth fine curve graphs and what is known about the associated subgroups of the homeomorphism group.

Nastaran Einabadi: Gromov classification of action of homeomorphisms on fine curve graphs
The homeomorphism group of a surface acts by isometries on its fine curve graph which is a Gromov hyperbolic space. These isometries can be classified into three types: elliptics, parabolics, and hyperbolics. It turns out that in the case of the torus, this classification is related to rotation sets, which capture the asymptotic direction and speed of orbits. For higher genus closed surfaces, hyperbolic homeomorphisms can be described as the ones having an ergodic homological rotation set with non-empty interior.
In this talk, I will present my results concerning the parabolic-elliptic distinction for higher genus surfaces. To do so, I will introduce a generalization of homotopic rotation sets adapted to capture finer information about the directions of trajectories of the dynamics, especially in the case where trajectories have sublinear speed.

Alejo García-Sassi: Models for hyperbolically-acting surface homeomorphisms
The idea of the talk is to help organize chaos in surface dynamics. We will study isotopic-to-the-identity homeomorphisms of closed surfaces whose ergodic rotation set has nonempty interior, which we know are those which act hyperbolically on the fine curve graph of the surface (by Bowden, Hensel, Mann, Militon and Webb in the torus case, and by Guihéneuf and Militon in the higher genus case).
In joint work with Fábio Tal, we show that every map with this property is monotonically semiconjugate to a homeomorphism of the same surface, with the same rotation set, which is fully chaotic: it preserves area, has a single homoclinic class, and has a dense family of rotational horseshoes.

Alejandro Kocsard: Forcing theory in Dehn homotopy classes of torus homeomorphisms
Forcing theory deals with the study of mechanisms that force complex and rich dynamics in a dynamical system. The celebrated Sharkovskii's theorem is a paradigmatic example of in this theory. There are some other well known results in forcing theory that are of homotopical nature, such as the forcing of periodic orbits and positive topological entropy for homeomorphisms of surfaces that are homotopic to Anosov or pseudo-Anosov homeomorphisms. In the last decade we have seen the development of a forcing theory for surface homeomorphisms in the identity homotopy class due to Le Calvez and Tal.
In this talk we will present some recent results on forcing theory for homeomorphisms of the torus that are homotopic to Dehn twists. As a consequence of our results, we show that there is no homeomorphism of the torus that is homotopic to a Dehn twist and that parabolically acts on the fine curve graph of the torus.
This is a joint work with Heric Corrêa (IMPA).

Emmanuel Militon: Invariant partitions for homeomorphisms isotopic to a pseudo-Anosov homeomorphism
In this talk, we will discuss how much the dynamics of a homeomorphism isotopic to a pseudo-Anosov homeomorphism looks like the dynamics of the actual pseudo-Anosov homeomorphism. More precisely, we will discuss the connections between a semi-conjugacy result by Handel, a more precise result by Fathi which gives invariant stable and unstable partitions of the surface and a new way to obtain these partitions.

Nelson Schuback: A Foliated Viewpoint on homotopy Brouwer Theory
Brouwer homeomorphisms are orientation-preserving, fixed-point-free homeomorphisms of the plane. In recent years, their dynamics have mainly been studied through two complementary approaches: one introduced by Handel and the other by Le Calvez, each providing a distinct perspective on their behavior. In this talk, we present a unified framework in which the foliated methods developed by Le Calvez recover, and in some cases extend, classical results from Handel’s theory.

Jing Tao: Tempered maps of surfaces of infinite type
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 tempered 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.

Nicholas Vlamis: Parabolic isometries via Denjoy blowups
Fanoni and Hensel constructed a homeomorphism of the torus that acts as a parabolic isometry of the fine curve graph and preserves a Cantor set in the torus. This allowed them to obtain a parabolic isometry of the non-separating curve graph of an infinite-type surface, answering a question about big mapping class groups. In this talk, I will present a new proof of their result using the Gromov boundary of the fine curve graph. The key ingredient is an extension of work of Bowden, Hensel, and Webb on bi-foliated structures on surfaces to their Denjoy blowups. This perspective yields parabolic isometries of the non-separating curve graph for higher-genus surfaces, extending the Fanoni–Hensel example beyond the torus. This is part of ongoing joint work with Carolyn Abbott, George Domat, and Priyam Patel.

Richard Webb: Equators of the two-sphere and area-preserving homeomorphisms
The curve graph is an important tool in the study of mapping class groups of surfaces, and several other areas of low-dimensional topology and geometry. It turns out that the homeomorphism group of a positive-genus closed surface can be studied using an analogous object now called the fine curve graph introduced by Bowden, Hensel, and the speaker. For instance, one can show that the identity component has non-vanishing scl. However, for the 2-sphere, it was already known that scl vanishes, so the fine curve graph machinery doesn't work for the homeomorphism group. In this talk, we introduce (uncountably many) analogues of the fine curve graph in order to study the group of area-preserving homeomorphisms (and/or Hamiltonian diffeomorphisms) of the 2-sphere. How this relates to the dynamics of area-preserving homeomorphisms, and symplectic geometry, is particularly curious. Using these new tools, we are able to construct new quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere, which are C^0 continuous and vanish on the stabiliser of the standard equator. I will discuss an application of this regarding the geometry of simple closed curves that separate the sphere into two components of equal area (i.e. equators of the sphere), and how this relates to the Equator Conjecture. Joint work with Yongsheng Jia.

Participants
Pedro Alves
Katherine Booth
Nastaran Einabadi
Hélène Eynard-Bontemps
Federica Fanoni
Sebastian Hensel
Vincent Humilière
Alejo García-Sassi
Pierre-Antoine Guihéneuf
Yongsheng Jia
Alejandro Kocsard
Frédéric Le Roux
Yusen Long
Kathryn Mann
Achille Méthivier
Emmanuel Militon
Alejandro Passeggi
Priyam Patel
Anna Ribelles Pérez
Martín Sambarino
Roberta Schapiro
Nelson Schuback
Jing Tao
Yvon Verberne
Nicholas Vlamis
Richard Webb

Organizers
Federica Fanoni
Sebastian Hensel
Frédéric Le Roux
Kathryn Mann

Sponsors
ANR Tremplin MAGIC
Institut de Mathématiques de Jussieu-Paris Rive Gauche