Schedule & Talk info

The conference will be held at the Room 302, A5 Building, Institute of Mathematics, Vietnam Academy of Science and Technology (IM VAST), 18B Hoang Quoc Viet Street, Cau Giay district, Hanoi, Vietnam

Ara Basmajian (City University of New York)

Title: From Collars on Riemann Surfaces to Tubes in Complex Ball Quotients

Abstract: The celebrated Keen collar lemma guarantees that a simple closed geodesic on a hyperbolic Riemann surface has a collar (tubular neighborhood) whose width only depends on its length. Viewing a Riemann surface as the quotient of the unit ball in the complex plane, a natural generalization is to ball quotients in higher dimensions where the Poincaré metric is replaced by the Bergman metric (also known as the complex hyperbolic metric). Such ball quotients are called complex hyperbolic manifolds. The focus of this talk will be on embedded complex geodesics in complex hyperbolic 2-manifolds; a complex geodesic has complex codimension one in the quotient complex 2-manifold. We prove a tubular neighborhood theorem for such a complex geodesic where the width of the tube depends only on the Euler characteristic of the embedded complex geodesic. We also derive an explicit estimate for this width. After giving a short history of the collar lemma generalizations and discussing the basics of complex hyperbolic geometry, we will discuss the ideas leading to the proof of this tubular neighborhood theorem. This is joint work with Youngju Kim.

Nhat Minh Doan (IM VAST and National University of Singapore)

Title: Finiteness of ortho-integral surfaces

Abstract: We study ortho-integral (OI) hyperbolic surfaces with totally geodesic boundaries, where each orthogeodesic has an integer cosh-length. We prove that on OI surfaces, the square of the trace of every closed geodesic is an integer. Consequently, there are only finitely many OI surfaces of genus g with n geodesic boundary components. Additionally, we provide a complete classification of OI pairs of pants and one-holed tori. This is a work in progress joint with Khanh Le.

Masaharu Ishikawa (Keio University)

Title: On 2-dimensional knots in the 4-sphere invariant under circle actions

Abstract: Circle actions on homotopy 4-spheres are classified by Montgomery, Yang and Fintushel. Later, Pao proved that such homotopy 4-spheres are the standard 4-sphere. A 2-sphere embedded in the 4-sphere that is invariant under a circle action is called a branched twist spin. Twist spun knots are simplest examples of branched twist spins. The image of exceptional orbits of the action together with the image of the fix point set constitutes a 1-knot in the orbit space. Presentations of the fundamental groups of the complements of branched twist spins are given by Plotnick. Recently, Fukuda studied invariants of branched twist spins obtained from representations of their fundamental groups. In this talk, we focus on the fundamental groups of the 3-orbifolds obtained from the 1-knots in the orbit space and use it to show that different 1-knots correspond to different branched twist spins in most cases. This is a joint work with Mizuki Fukuda in Tohoku University.

Sang-hyun Kim (KIAS)

Title: Smoothing one-dimensional actions of countable groups with subexponential growth

Abstract: We prove that if a countable group does not contain a finitely generated subgroup of exponential growth, then every topological action of the group on a compact connected one-manifold can be blown-up to a C1 action. The proof is based on a functional characterisation of such groups. This is joint work with Nicolás Matte Bon, Mikael de la Salle and Michele Triestino.

Thomas Koberda (University of Virginia)

Title: Locally approximating groups of homeomorphisms of manifolds

Abstract: I will survey the model theory of locally approximating groups of homeomorphisms of compact manifolds, which are groups of homeomorphisms which are "sufficiently dense" in the full group of homeomorphisms, with the compact-open topology. These groups always interpret first-order arithmetic; using arithmetic, one can prove that all finitely generated subgroups of locally approximating groups are definable, with parameters. Under some further conditions, one can prove that these groups are prime models of their theories. I will also discuss action rigidity for these groups: if an arbitrary group G is elementarily equivalent to a locally approximating group of homeomorphisms of a compact manifold M, then for any locally approximating group action of G on a manifold N, we must have that M and N are homotopy equivalent to each other. In low dimensions, we may in fact conclude that M and N are homeomorphic to each other. This represents joint work with J. De la Nuez Gonzalez.

Sanghoon Kwak (KIAS)

Title: Mapping class groups of Infinite graphs -- Big Out(Fn)

Abstract: Surfaces and graphs are closely related; there are many parallels between the mapping class groups of finite-type surfaces and finite graphs, where the mapping class group of a finite graph is the outer automorphism group of a free group of (finite) rank. A recent surge of interest in infinite-type surfaces and their mapping class groups begs a natural question: What is the mapping class group of an “infinite” graph? In this talk, I will explain the answer given by Algom-Kfir and Bestvina, and present recent work, joint with George Domat and Hannah Hoganson, on the coarse geometry of such groups.

Khanh Le (Rice University)

Title: Order-preserving outer automorphisms of free and surface groups

Abstract: A group is bi-orderable if it admits a total ordering that is left and right invariant. Orderable groups have received recent attention due to their connection with dynamical group theory and with 3-manifold groups via the L-space conjecture. Given a bi-orderable group G, it is natural to ask which outer automorphisms of G preserve a bi-ordering on G since these correspond precisely to cyclic extension of G that is bi-orderable. Motivated by the connection with 3-manifolds, we focus on the case when G is a free group or a bi-orderable surface group. In this talk, I will also describe a criterion for an outer automorphism of a free group induced by a braid action to be order-preserving using the reduced Burau representation. This is a work in progress joint with Jonathan Johnson.

Hidetoshi Masai (Musashino Art University)

Title: On distances and horofunctions on the Teichmuller space

Abstract: We first discuss a general procedure to generate distances and horofunctions via a function of type F: X\times Y\to \R. Then, in this talk, we focus on the Extremal length functional extended on the space of geodesic currents by Martínez-Granado and D. Thurston. We then construct a distance on the Teichmuller space which is a sibling of the Teichmuller metric.

Manh-Tien Nguyen (University of Luxembourg)

Title: CMC foliation of hyperbolic 3-manifolds and Hamiltonian flow on the cotangent of Teichmüller space

Abstract: It was conjectured by Thurston that certain hyperbolic 3-manifolds can be foliated by surfaces of constant mean curvature (CMC). By keeping track of the conformal structure and the trace-free part of the second fundamental form of each leave, one can view such foliation as a flow on the cotangent of Teichmüller space, which was known (Moncrief 1989) to be Hamiltonian under the canonical symplectic structure. In this work, joint with F. Mazzoli, J.-M. Schlenker and A. Seppi, we establish long-time existence of the flow and confirm the conjecture for a class of 3-manifolds.

Hoang Thanh Nguyen (Danang FPT University)

Title: Quasi-redirecting boundaries of non-positively curved groups

Abstract: Qing-Rafi recently introduced a new boundary for metric spaces called the quasi-redirecting (QR) boundary. The QR boundary is invariant under quasi-isometries, is often compact, and contains sublinearly Morse boundaries as topological subspaces. However, it is unknown whether all finitely generated groups have well-defined QR boundaries. In this talk, we demonstrate that the quasi-redirecting boundary is well-defined as a topological space for several groups of nonpositive curvature. Additionally, we provide a complete description of the QR boundaries for admissible groups that act geometrically on CAT(0) spaces. This is joint work with Alex Margolis and Yulan Qing.

Yulan Qing (University of Tennessee)

Title: Genericity in groups via boundary

Abstract:  In this talk we will discuss the behavior of sublinearly Morse directions under first passage percolation. We first recall the construction of the sublinearly Morse boundary, which is a metrizable and QI-invariant topological space that enjoys some genericity properties. First passage percolation is a model of random perturbation of a given geometry. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph has bounded degree and contains a sublinearly Morse geodesic, then almost surely, there exists a bi-infinite geodesic in first passage percolation on the graph.  This work in progress joint with Sagnik Jana.

Jenya Sapir (Binghamton University)

Title: Geometry of geodesic currents

Abstract: The space of projective, filling currents $\mathbb P \mathcal C_{fill}(S)$ contains many structures relating to a closed, genus $g$ surface $S$. For example, it contains the set of all closed curves on $S$, as well as an embedded copy of Teichm\"uller space, and many other spaces of metrics on $S$. It turns out that the Thurston metric on Teichm\"uller space extends to $\mathbb P \mathcal C_{fill}(S)$. We will discuss the geometry of $\mathbb P \mathcal C_{fill}(S)$ with this metric.

Xiaobing Sheng (Okinawa Institute of Science and Technology)

Title: Thompson knot theory and the conjugacy classes of $F$

Abstract: Jones found a concrete way to construct knots and links from elements of Thompson’s group $F$ which is an interesting finite presented infinite group having many counter-intuitive properties. Aiello has summarised the program as Thompson knot theory.

Many properties of $F$ from the viewpoint of classical group theory were much investigated. The conjugacy problem of $F$ has been solved by Guba and Sapir and lately by Belk and Matucci from a more dynamical perspective by using the so-called strand diagrams.

In an attempt to tackle the Markov theorem of $F,$ we found that there could be an interesting relation between conjugacy classes of $F$ and Thompson knot theory by considering annular strand diagrams related to the group elements and we proved that for any knot or link, there exist elements from infinitely many conjugacy classes of Thompson’s group $F$ that realises it via Jones' construction. This is a joint work with Yuanyuan Bao.

Hanh Vo (Arizona State University)

Title: Short geodesics with self-intersections

Abstract: We consider the set of closed geodesics on a hyperbolic surface. Given any non-negative integer k, we are interested in the set of primitive essential closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.

Binbin Xu (Nankai University)

Title: Equivalent Curves on Surfaces

Abstract: We consider a closed oriented surface of genus at least 2. To describe curves on it, one natural idea is to choose once for all a collection of curves as a reference system and to hope that any other curve can be determined by its intersection numbers with reference curves. For simple curves, using the work of Dehn and Thurston, it is possible to find such a reference system consisting of finitely many simple curves. The situation becomes more complicated when curves have self-intersections. In particular, for any non negative integer k, it is possible to find a pair of curves having the same intersection number with every curve with k self-intersections. Such a pair of curves are called k-equivalent curves. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k's. This is a joint-work with Hugo Parlier.

Abdul Zalloum (Institute for Advanced Study in Mathematics of HIT)

Title: Producing actions on hyperbolic spaces from walls.

Abstract: Starting with an arbitrary set S and a collection of walls W (bi-partitions of S), Sageev's construction allows us to produce a CAT(0) cube complex X; assuming that only finitely many walls separate a pair of points in S. If G acts on S preserving W, then G acts on the resulting CAT(0) cube complex. 

I will discuss a generalization of Sageev's construction that allows one to produce actions on hyperbolic (and injective) metric spaces, starting with a set S, a collection of walls W that satisfy a weaker finiteness property compared to Sageev's. Time permitting, I will discuss applications. This is joint with Petyt-Spriano.

Alexis Marchand (Kyoto University)

Title: Stable commutator length, surfaces, and rationality

Abstract. Stable commutator length (scl) is a measure of the homological complexity of group elements, which has attracted attention for its connections with notions of negative curvature in geometric group theory, such as Gromov-hyperbolicity. I will introduce scl, with a focus on algorithmic computations and rationality problems. I will present some results (joint with Henry Wilton) aiming to make progress towards understanding scl in surface groups.

Changjie Chen (CRM and Université de Montréal)

Title: Morse theory on the moduli spaces

Abstract: I will introduce some Morse functions on the moduli space of Riemann surfaces and talk about the associated Morse theory with conclusions on the homology of the moduli spaces.

Minh Hien Huynh (Quy Nhon University)

Title: A quantitative closing lemma on factors of the hyperbolic plane

Abstract: In this talk we introduce a quantitative version of the closing lemma for factors of the hyperbolic plane. As an application we study partner geodesics for closed geodesics having encounters.

Ryo Matsuda (Kyoto University)

Title: On David map and Teichmüller spaces of infinite type Riemann surfaces

Abstract: The Teichmüller space is the moduli space of Riemann surfaces with markings by quasiconformal mappings. Thus, it is natural to consider degenerate quasiconformal mappings when constructing objects that belong to its boundary. In this talk, we will introduce a construction of David maps, a type of degenerate quasiconformal mapping, that are equivariant with respect to infinitely generated Fuchsian groups. Furthermore, we will prove that the corresponding Kleinian group lies on the Bers boundary; that is, we will construct a Kleinian group whose discontinuity domains have two connected components and appear on the Bers boundary. If time permits, we will also discuss constructing a holomorphic family of such Kleinian groups parametrized by open subsets of infinite-dimensional Banach space.

Inyoung Ryu (Texas A&M University)

Title: Connected components of spaces of type-preserving representations

Abstract: We investigate the spaces of representations of surface groups into PSL(2, R). For a closed surface, by the classic result of Goldman, the Euler class together with the Milnor-Wood inequality provide a complete classification of the connected components of the spaces of the representations. However, describing the connected components becomes more subtle when considering the space of type-preserving representations for punctured surfaces. In this talk, I will present a recent joint work with Tian Yang that addresses this problem. 

Ivan Telpukhovskiy (YMSC, Tsinghua University)

Title: On canonical ideal cell decompositions of decorated hyperbolic surfaces.

Abstract: We give another proof of the following folkloric result: the Epstein-Penner ideal cell decomposition of a decorated hyperbolic surface with punctures matches the Bowditch-Epstein ideal cell decomposition. Further, we introduce a generalized notion of the cut locus (also known as spine or collision locus) for crowned hyperbolic surfaces, and show that we naturally get two distinct decompositions of decorated Teichmüller space of surfaces with crowns. Based on a joint work in progress with Yi Huang.

Quoc Bao Vo (IM VAST)

Title: Pro-Algebraic Completion of Surface Groups

Abstract: Fix a field k. Given a discrete group G, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group G^{alg} that is universal with respect to finite-dimensional representations of G. For k =\mathbb{C}, the talk presents the use of pro-algebraic completion to compute the differential fundamental group π(X/k) and proves that the cohomology of the differential fundamental group π(X/k) is isomorphic to the de Rham cohomology when X is a projective curve of genus g > 0.

Sangsan (Tee) Warakkagun (Khon Kaen University)

Title: Pólya's Shire Theorem for Riemann Surfaces

Abstract: Pólya's classical Shire Theorem states that the zeros of the successive derivatives of a meromorphic function on the complex plane accumulate onto the edges of the Voronoi diagram determined by the loci of the poles of the function. We develop a generalization to describe the limit set of the zeros of the iterates of a meromorphic function on a compact Riemann surface under a linear differential operator defined by a meromorphic 1-form. Refining Pólya's local arguments, we show that the accumulation set is the union of the edges of a generalized Voronoi diagram defined by the meromorphic function and the singular flat metric induced by the 1-form. This is an ongoing work in progress with Rikard Bögvad, Boris Shapiro, and Guillaume Tahar.