Title/Abstracts

09:30 - 10:10 

Yi Liu (Peking University)

Title: On profinite properties of 3-manifolds 

Abstract: In this talk, I plan to review recent progress on profinite completion of 3-manifold groups. I will survey on what is known and what is unknown yet, and explain some recent work of mine regarding profinite invariance/non-invariance of Turaev Viro invariants. 

10:20 - 11:00 

Kento Sakai (Osaka University)

Title: On families of hyperbolic surface via Teichmüller theory of harmonic maps, and their limits 

Abstract: In the Teichmüller theory of harmonic maps, we obtain a certain parametrization of Teichmüller space. Through the parametrization, a ray in the vector space of quadratic differentials determines this family, so the family is called a harmonic map ray. In the case of finite type surfaces, the degeneration along harmonic map ray can be described from the viewpoint of Thurston’s boundary. The main problem of this talk is, in the case that surfaces have crown-like boundaries, what is the limit of a harmonic map ray? In this talk, for the hyperbolic ideal polygons, we formulate the limit in the Gromov-Hausdorff sense. Our result is based on the result of Gupta’s parametrization of Teichmüller space.

11:30 - 11:50

Virginie Charette (University of Sherbrooke)

Title : The geometry of the bidisk

Abstract : The bidisk is the product of two copies of the hyperbolic plane. Its geometry has some curious features, which we will discuss in this talk.

12:00 - 12:20

Youngju Kim (Konkuk University)

Title: On complex hyperbolic manifolds

Abstract: We will discuss basic properties of complex hyperbolic manifolds. In particular, we focus on embedded totally geodesic subsurfaces. We prove a tubular neighborhood theorem for an embedded complex totally geodesic surface in a complex hyperbolic 2-manifold. The width of the tube depends only on the Euler characteristic of the embedded surface.

14:00 - 14:20

Kyeongro Kim (Seoul National University)

Title : Circle actions of 3 manifolds

Abstract : The foliations theory has played an important role in three-manifold theory. In particular, Thurston's study of mapping tori is one of the most important parts of geometric topology. He attempted to generalize the results to the case of tautly foliated three-manifolds. As a first step, He proved the universal circle theorem. In this talk, I will provide the context of the study of taut foliations and introduce Thurston's Universal Circle Theorem. Finally, I will discuss laminar groups and recent results. This talk is based on works with Harry Hyungryul baik and Hongtaek Jung.   

14:30 - 14:50

Hyungryul Baik (Korea Advanced Institute of Science and Technology)

Title: some open problems 

Abstract: I will discuss some open problems which arose from my research and explain the context. 

15:30 - 15:50

Jan Kim (Pusan National University)

Title: Non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups 

Abstract: In 1987, Gromov introduced the concept of hyperbolic groups, and formulated relatively hyperbolic groups as a generalization of hyperbolic groups. Very recently, Reinfeldt-Weidmann and Fujiwara-Sela proved that every hyperbolic group is Hopfian. On the other hand, in relation to the Hopf property for relatively hyperbolic groups, Osin posed a question of whether all relatively hyperbolic groups are Hopfian when their peripheral subgroups are Hopfian. In this talk, I will introduce several non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups that we constructed. Additionally, we will discuss a general construction method, referred to as the image extension theorem, to generate non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups. This is based on joint work with Arap kyzy Tattybubu and Donghi Lee.

16:00 - 16:20

Junseok Kim (Korea Advanced Institute of Science and Technology)

Title: A subgroup of Out(RAAG) and its acylindrical hyperbolicity

Abstract: In this talk, I will introduce a finite index normal subgroup of the outer automorphism group of right-angled Artin group generated by transvections and partial conjugations. In order to check whether this group is acylindrically hyperbolic or not, we will discuss some interesting algebraic structure of it when the defining graph of right-angled Artin group is connected.

16:30 - 16:50

Dongsoo Lee (Korea Advanced Institute of Science and Technology)

Title: On the group of homology $S^1 \times S^2$'s

Abstract: Kawauchi defined a group $\Omega(S^1 \times S^2)$ on the set of homology $S^1 \times S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. We will talk about the kernel of the zero-surgery homomorphism and the $2$-torsion subgroup of $\Omega(S^1 \times S^2)$.

09:30 - 10:10 

Wenyuan Yang (Peking University)

Title: Limit sets for branching random walks on relatively hyperbolic groups  

Abstract: Branching random walks (BRW) on groups consist of two independent processes on the Cayley graphs: branching and movement. Start with a particle on a favorite location of the graph. According to a given offspring distribution, the particles at the time n split into a random set of particles with mean $r \ge 1$, each of which then moves independently with a fixed step distribution to the next locations. It is well-known that if the offspring mean $r$ is less than the spectral radius of the underlying random walk, then BRW is transient: the particles are eventually free on any finite set of locations. The particles trace a random subgraph which accumulates to a random subset called limit set in a boundary of the graph. In this talk, we consider BRW on relatively hyperbolic groups and study the limit set of the trace at the Bowditch and Floyd boundaries. In particular, the Hausdorff dimension of the limit set will be computed. This is based on a joint work with Mathieu Dussaule and Longmin Wang. 

10:20 - 11:00 

Hidetoshi Masai (Tokyo Tech)

Title: On iso-orthospectrum surfaces. 

Abstract: For a surface with totally geodesic boundary, an orthogeodesic is a geodesic whose endpoints are perpendicular to the boundary. The orthospectrum is a set of lengths of orthogeodesics. There are interesting identities due to Basmajian, Bridgeman, and others with respect to orthogeodesics. I will first quickly review known results about orthospectrum and then talk about iso-orthospectrum surfaces. This is based on the joint work with Greg McShane. 

11:30 - 11:50

Inhyeok Choi (Korea Institute For Advanced Study)

Title: Regularity of the escape rate and the asymptotic entropy of a random walk

Abstract: Let G be a countable group acting on a metric space X. One popular method of sampling random elements of G is to observe random walks on G. Given a random walk on G there are two asymptotic quantities, called the escape rate and the asymptotic entropy, that tell us how quickly the random walk evolves on X and in G, respectively. In this talk, I will summarize known results about regularity of these quantities with respect to the underlying probability measure, and report a new results, some of which is independently due to Anna Erschler and Joshua Frisch.

12:00 - 12:20

Ser-wei Fu(National Taiwan University)

Title: Flat Grafting Deformation

Abstract: In this talk the goal is to explore deformations to hyperbolic and flat surfaces. I will try to provide an overview of interesting properties that require closer looks. Finally I will introduce flat grafting and discuss its properties.

09:30 - 10:10 

Shinpei Baba (Osaka University)

Title: Bending Teichmüller spaces and character varieties. 

Abstract: The boundary of the convex core of a hyperbolic three-manifold is, as Thurston found, a hyperbolic surface bent along a measured lamination. By lifting it to the universal cover, we see the hyperbolic plane bent in the hyperbolic 3-space along a measured lamination in an equivariant manner. This equivariant property yields a holonomy representation of the fundamental group of the surface into the isometry group of the 3-space. By bending the homeomorphic hyperbolic surfaces along a fixed measured lamination, we obtained a mapping of the Teichmüller space into the space of such representations. We discuss some interesting properties of this mapping between the deformation spaces which are similar to some properties of the equivariantly bent hyperbolic plane. 

10:20 - 11:00 

Yi Huang (Yau Mathematical Sciences Center)

Title: The idiot's guide to shearing surfaces 

Abstract: we (try to) give a friendly guide for shearing between hyperbolic surfaces in as ``efficient" a manner as possible. On the way, we'll see Teichmueller spaces, Thurston’s earthquake theorem, and a novel metric on Teichmueller space called the earthquake metric which has surprising connections to both the Thurston metric and the Weil-Petersson metric. This is work in collaboration with K. Ohshika, H. Pan and A. Papadopoulos. 

11:30 - 11:50

Donggyun Seo (Seoul National University)

Title: Comparison between two length spectra of the genus two handlebody group 

Abstract: A mapping class group of a handlebody, known as a handlebody group, is not just a significant object in low-dimensional topology but also a fascinating subject in geometric group theory. To analyze the geometry of a handlebody group, we commonly inspect two metric spaces where the handlebody group acts. One is a Teichmüller space, and the other is a Culler-Vogtmann outer space. Each action is naturally derived, but no canonical way to compare their properties exists. In this talk, we will see how different the translation lengths with those actions are, especially for the genus two handlebody group. This is joint work with KyeongRo Kim. 

12:00 - 12:20

Heejoung Kim (Kyungpook National University)

Title: Conformal dimension of the boundary for a hyperbolic group

Abstract: The Gromov boundary of a hyperbolic group is one of the useful and fundamental tools for studying the group. Even if the Gromov boundary of a hyperbolic group is metrizable, the metric is not canonical. In 1989, Pansu introduced the conformal dimension of the Gromov boundary, which is a quasi-isometry invariant by construction. In general, the conformal dimension is difficult to compute or even estimate. In this talk, we discuss the conformal dimension of the boundary for a hyperbolic group. In particular, we talk about the boundary for a hyperbolic Coxeter group and how to compute its conformal dimension, which is an ongoing joint project with Dymarz, Heller, Vo, and Thoma.

14:00 - 14:20

Juhun Baik (Korea Advanced Institute of Science and Technology)

Title: Topological normal generation of big mapping class groups

Abstract: By Lanier and Margalit, there are many mapping classes for finite type closed surfaces with genus more than 3, whose normal closure is the whole mapping class group. In other words, the mapping class groups are normally generated. We ask whether the mapping class group of an infinite type surface is also normally generated or not. In this talk, I will introduce some properties of mapping class groups of infinite type surfaces and explain some open questions.

14:30 - 14:50

Wonyong Jang (Korea Advanced Institute of Science and Technology)

Title: On the kernel of group actions on asymptotic cones 

Abstract: The concept of an asymptotic cone was first suggested by Gromov and he used it to establish Gromov's polynomial growth theorem. An asymptotic cone of a group reflects many properties of the group. For example, a group is virtually nilpotent if and only if all of its asymptotic cones are locally compact (equivalently, proper). Also, a finitely generated group is hyperbolic if and only if all of its asymptotic cones are real trees.

 In this talk, we characterize the natural kernel of the action of a group G on its asymptotic cone. Our main theorem states that if G is acylindrically hyperbolic, then the kernel of G-action on an asymptotic cone of G is the same as many algebraically defined subgroups. Moreover, this result does not depend on the choice of ultrafilter and sequence that we need to define asymptotic cones so it implies that the kernel is invariant under the choice of these. It is known that a group may have distinct (actually, non-homeomorphic) asymptotic cones, and indeed some acylindrically hyperbolic groups also have various asymptotic cones.

 We also relate this kernel to other kernels of group actions on other spaces at "infinity", for instance, the limit set of convergence group action, Floyd boundary, and many boundaries of CAT(0) spaces with some conditions. If time permits, we will introduce another action of G on an asymptotic cone, called Paulin's construction, and describe the kernel of Paulin's construction. This work is joint with my advisor, Hyungryul Baik.

15:30 - 15:50

Javier de la Nuez Gonzaléz (Korea Institute For Advanced Study)

Title: Minimality of the compact-open topology on diffeomorphism and homeomorphism groups 

Abstract: We will talk about recent work in which we prove that the restriction of the compact-open topology to the diffeomorphism group of a manifold without boundary of dimension different from 3 is a minimal element of the lattice of Hausdorff group topologies on the group. If the dimension is also different from 4 it follows that the same holds for the compact-open topology on the homeomophism group, which combined with K. Mann's automatic continuity results implies the latter admits a unique separable Hausdorff group topology.

16:00 - 16:20

Minkyu Kim (Korea Institute For Advanced Study)

Title: On analytic exponential functors from free groups.

Abstract: Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. There are examples given by Hochschild-Pirashvili homology for wedge of circles, or functors on the Habiro-Massuyeau category of Jacobi diagrams in handlebodies; as another example, bivariant functors on gr provide natural coefficients for the stable cohomology of automorphism groups of free groups. Some of these natural examples satisfy further properties: they are analytic and/or exponential.

Pirashvili proves that the category of exponential contravariant functors from gr to the category k-Mod of k-modules is equivalent to the category of cocommutative Hopf algebras over k. Powell proves an equivalence between the category of analytic contravariant functors from gr to k-Mod, and the category of linear functors on the linear PROP associated to the Lie operad when k is a field of characteristic 0. In this talk, after explaining these two equivalences of categories, I will explain how they interact with each other. We also go further by introducing primitive contravariant functors on gr to extend the results to positive characteristic. This is a joint work with Christine Vespa.

09:30 - 10:10 

Chenxi Wu (University of Wisconsin-Madison)

Title: End periodic maps on graphs

Abstract: I will discuss some examples and properties of end periodic mapping classes of infinite graphs, and discuss their potential applications to geometric group theory and tropical geometry.

10:20 - 11:00 

Sanghyun Kim (Korea Institute For Advanced Study)

Title: First order rigidity of manifold homeomorphism groups 

Abstract: Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker’s theorem on isomorphic homeomorphism groups (1963) without relying on it. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS). 

11:30 - 12:10

Anna Rois Sanchis (Sorbonne Université)

Title:  On the length spectrum of random hyperbolic 3-manifolds

Abstract: We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model.