Mini-courses
Harry Baik (KAIST, Daejeon)
[Part I-III]
Title: Normal Generators of Mapping Class Groups
Abstract: We will begin by reviewing the theory of Lanier-Margalit, which provides various criteria for a given mapping class to be a normal generator of the mapping class group. A key consequence of this theory is that a pseudo-Anosov element with a small translation length on the Teichmüller space is a normal generator. We will then explore two possible generalizations: one is to replace pseudo-Anosov elements with reducible elements, and the other is to replace the Teichmüller space with the curve graph.
Sanghoon Kwak (Seoul National University)
[Part I]
Title: Big Out(Fₙ) and its algebraic properties
Abstract: The group Out(Fₙ) consists of automorphisms of the free group of rank n, modulo inner automorphisms, and is regarded as the mapping class group of finite graphs. Algom-Kfir and Bestvina introduced “Big Out(Fₙ)” as the mapping class group of (locally finite) infinite graphs. In this talk, I will give a gentle introduction to Big Out(Fₙ) and its various interesting algebraic properties. This is joint work with George Domat and Hannah Hoganson.
[Part II]
Title: Big Out(Fₙ) and its coarse geometry
Abstract: As a geometric group theorist, one might be also interested in studying Big Out(Fₙ) as a metric space. One typical way for an arbitrary group is to use its generating set and define the metric using word length. It is a classical result that a finitely generated or a compactly generated group has a well-defined metric regardless of the choice of the generating set. However, Big Out(Fₙ) is not even locally compact, so fails to be in a classical context. In this talk, I will introduce Rosendal’s remedy by extending the framework beyond compactly generated groups. Then we will see which Big Out(Fₙ) admits a well-defined coarse geometry under his frame. This is joint work with George Domat and Hannah Hoganson.
[Part III]
Title: ‘Mid’ Out(Fₙ) and its relatives in dimension 2 and 3
Abstract: In this talk, we consider the “asymptotically rigid mapping class group of infinite graphs” as a countable subgroup of Big Out(Fₙ), which has nice finiteness properties but at the same time contains Aut(Fₙ) as subgroups for all n. Hence, the subgroup nicely sits as ‘mid Out(Fₙ)’ between usual (small) Out(Fₙ) and Big Out(Fₙ). We will focus on the case when the graph has finitely many ends, and introduce the similar groups for surfaces and (doubled) handle bodies. We confirm that the graph ‘mid Out (Fₙ)’ are genuinely new class of groups in this industry; they are not commensurable to its surface and 3-manifold relatives. This is joint work with Thomas Hill, Brian Udall and Jeremy West.
Andrew Putman (University of Notre Dame)
[Part I-III]
Title: The topology of the mapping class group and its subgroups
Abstract: I will discuss finiteness results about the topology of the mapping class group, and in particular my recent work with Dan Minahan in which we compute the second rational cohomology group of the Torelli group.
Invited talks
Inhyeok Choi (KIAS)
Title: Locally finite invariant measures on measured laminations
Abstract: Let S be a hyperbolic surface. The space of measured laminations carries a natural locally finite MCG(S)-invariant measures called the Thurston measure. Lindenstrauss and Mirzakhani, and Hamenstädt independently proved that this is the unique locally finite invariant measure that is fully supported on uniquely ergodic laminations, or the “conical limit points” of MCG(S). I will explain a generalization of this result for subgroups of MCG(S). Our method does not make use of the mixing/recurrence property of the Teichmüller flow or the unipotent flow. Joint work with Dongryul M. Kim.
Koji Fujiwara (OIST/Kyoto University)
Title: Uniform exponential growth of discrete subgroups on Hadamard manifolds
Abstract: Let G be a finitely generated discrete subgroup of the isometry group of an Hadamard manifold of dimension n, whose sectional curvature is bounded between two negative constants a and b. It is known that there exists a constant C(n,a,b)>1 such that the growth rate of G is at least C.
In this work, we improve this result by giving a sharper lower bound on the growth rate, which takes into account the size of a generating set of G. We also discuss an application of this estimate.
This is joint work with Emmanuel Breuillard.
Yusen Long (Paris-Est Créteil)
Title: Topological properties of the boundary of fine curve graph
Abstract: In 2022, Bowden, Hensel, and Webb introduced the fine curve graph to study the homeomorphism group of surfaces. Notably, they showed that this graph is Gromov hyperbolic. In this talk, I will discuss the topology of its Gromov boundary for closed surfaces of higher genus, focusing on properties such as connectedness and non-compactness. These results rely on the bounded geodesic image theorem for the fine curve graph. This work is joint with Dong Tan.
Masato Mimura (Tohoku University)
Title: An invitation to invariant quasimorphisms
Abstract: Given a group-normal-subgroup pair (G,N), we can define the notion of G-invariant (homogeneous) quasimorphisms on N. We will give an overview on recent developments around this concept, including applications on symplectic geometry and mapping tori of a surface. Based on series of joint work with Morimichi Kawasaki (Hokkaido), Mitsuaki Kimura (Osaka Dental), Shuhei Maruyama (Kanazawa) and Takahiro Matsushita (Shinshu).
Rachel Skipper (University of Utah)
Title: Braiding groups of homeomorphisms of the Cantor set
Abstract: In this talk we will discuss some recent work on groups which connect self-similar and Higman-Thompson groups to big mapping class groups via "braiding". We will explain some results on the topological finiteness properties of the resulting groups, which are topological generalizations of the algebraic properties of being finitely generated and finitely presented. The talk will involve recent joint works with Xiaolei Wu (Fudan) and Matthew Zaremsky (Albany).
Wenyuan Yang (BICMR, Beijing)
Title: Dimensions of escaping and recurrent geodesics
Abstract: In this talk, we investigate the asymptotic behavior of geodesics—such as escaping and recurrent trajectories—on Riemannian manifolds. Recurrent geodesics are characterized by their endpoints in the visual boundary, which correspond to conical limit points for the fundamental group. The Hausdorff dimension of (uniformly) conical points has been extensively studied, beginning with Patterson’s work (1975) on Fuchsian groups, followed by Sullivan (1979) for geometrically finite Kleinian groups, and later Bishop-Jones (1996) for general Kleinian groups. In this talk, we extend these classical results by computing the Hausdorff dimensions of two other key subsets of the limit set: The Myrberg limit set (a distinguished subclass of non-uniformly conical points) and the non-conical limit set. This is based on joint work with Mahan Mj (TIFR).