2K-GATE Workshop I

June 10-13, 2025

Seoul, Korea

Title and Abstracts

June 10th (Tue)

9:30 - 10:30Khanh Le (Rice University)Title: Arithmetic aspects of ortho-length spectrumAbstract: For a hyperbolic surface X with non-empty boundary, the ortho-length spectrum of X is the multi-set of lengths of (ortho)-geodesic arcs meeting the boundary of X perpendicularly at both endpoints. The spectrum was introduced by Basmajian for hyperbolic n-manifolds in the context of studying co-dimensional one embedded geodesic submanifolds. The ortho-length spectrum has been studied extensively in relation to geometric identities, hyperbolic geometry of surfaces and inverse geometry. In this work, we initiate a new investigation into the arithmetic properties of the ortho-length spectrum. In particular, we consider hyperbolic surfaces whose cosh-of-length of ortho-geodesics are all integers, ortho-integral surfaces. We prove that there are only finitely many ortho-integral surfaces for a fixed topological type. We also constructed infinitely many pairwise non-commensurable ortho-integral surfaces. In the talk, I will motivate the consideration of such surfaces and discuss some ideas and tools going into the proof. If time permits, I will also discuss some further questions that we would like to investigate. This is joint work with Minh Doan.

11:00 - 12:00Jason Behrstock (CUNY)Title: Hierarchically hyperbolic groups: an introductionAbstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, most cubulated groups, and others.In this talk I'll provide an introduction to studying groups and spaces from this point of view, both describing new tools to use to study these groups and applications of those results.

1:30 - 2:30Bruno Martelli (Università di Pisa)Title: A 4-dimensional pseudo-anosov homeomorphismAbstract: A celebrated theorem of Thurston states that a 3-manifold that fibers over the circle has a hyperbolic structure if and only if the monodromy is pseudo-Anosov. It is natural to wonder whether some version of this theorem holds in higher dimension. We will show that a natural strong version of this theorem holds at least in one case: there is a hyperbolic 5-manifold that fibers over the circle, and the monodromy is pseudo-Anosov in some very natural sense.

3:00 - Lightning talk session

Hyein Choi (Rice University)
Title: Quasi-isometric embeddings of finite quotients of Euclidean buildings
Hongtaek Jung (Seoul National University)
Title: Generic properties of Hitchin representations
Insung Park (Stony Brook University)
Title: Cube complexes of pre-homotopic mutlicurves and puncture-forgetting from measured foliations
Inyoung Ryu (Texas A&M University)
Title: Connected components of spaces of type-preserving representations
Donggyun Seo (Seoul National University)
Title: On the length space of free splittings of a free group

June 11th (Wed)

09:30 - 10:30Hoang Thanh Nguyen (FPT University)Title: Quasi-redirecting boundaries of groups.Abstract: Qing and Rafi recently introduced a new boundary for metric spaces, called the quasi-redirecting (QR) boundary. This boundary is quasi-isometry invariant, often compact, and contains the sublinearly Morse boundary as a topological subspace. While the existence of the QR boundary for all finitely generated groups remains an open question, we establish well-defined QR boundaries for several well-studied classes of groups, including relatively hyperbolic groups and all finitely generated 3-manifold groups.
We also demonstrate a connection between the QR boundary and the divergence of groups: groups with linear divergence have single-point QR boundaries, whereas certain groups with quadratic divergence, such as graph manifolds and CAT(0) admissible groups, have QR posets of height 2. Some open questions will be discussed if time permits.
This talk is based on joint work with Yulan Qing.

11:00 - 12:00Jason Behrstock (CUNY)Title: Quasiflats and quasi-isometric rigidity in hierarchically hyperbolic spacesAbstract: Quasi-isometric embeddings of euclidean space are a valuable tool for understanding groups and metric spaces. In this talk I'll show how the perspective of hierarchical hyperbolicity provides a way to classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. In particular, I will discuss how this study of quasiflats can be used to prove quasi-isometric rigidity for certain families of groups. This is joint work with Mark Hagen and Alessandro Sisto.

June 12th (Thu)

09:30 - 10:30Lvzhou ChenTitle: Maximal rotation and stable commutator lengthAbstract: The rotation number rot(h) of an orientation-preserving homeomorphism h of the circle measures how fast h rotates, and it captures the dynamical behavior of h. For a free group F acting on the circle and a word w in the commutator subgroup of F, rot(w) is a well-defined real number. The maximal rotation number maxrot(w) is the maximum of rot(w) over all F actions on the circle. When w is the standard relator of a surface group, this is well understood in Milnor-Wood inequalities. In general, there is an upper bound via Bavard's duality in terms of scl(w), the stable commutator length of w, which is a relative Gromov norm and roughly speaking measures the minimal complexity of surfaces bounding w. We will give some first examples of w for which the inequality maxrot(w)<=2scl(w) is not sharp, explain an algorithm that computes the integer part of maxrot(w), and discuss several unsolved problems. This is joint work with Geoffrey Baring, Danny Calegari and Alden Walker.

11:00 - 12:00Christian Rosendal (University of Maryland - College Park)Title: Aspects of automatic continuity 1Abstract: A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and refinements of the closed graph theorem, applying these to the problem of continuity of universally measurable homomorphisms and also to gauge the amount of choice needed to construct discontinuous homomorphisms between Polish groups.

1:30 - 2:30David Xu (KIAS)Title: Discrete isometry groups of the infinite-dimensional hyperbolic spaceAbstract: Similarly to Euclidean spaces, there is an infinite-dimensional analog for the (algebraic) hyperbolic spaces. This space enjoys all the "geometric" properties of its finite-dimensional siblings. However, the topological aspects of this space and its group of isometries are more involved than in finite dimension. In particular, even the notion of discrete isometry groups needs to be specified in this context. In this talk, I will present the infinite-dimensional hyperbolic space and describe some of its properties, emphasizing some differences with finite dimensions. Then, I will discuss some ways of finding "discrete" subgroups of isometries by bending convex-cocompact representations of surface groups arising from the exotic representations of Monod and Py.

2:45 - 3:45 (Note the time)Koichi Oyakawa (Vanderbilt University)Title: Geometry and dynamics of the extension graph of graph product of groupsAbstract: In this talk, I will introduce the extension graph of graph product of groups and explain its geometry. This notion enables us to study the properties of graph product by exploiting the large-scale geometry of its defining graph. In particular, I show that the asymptotic dimension of the extension graph exhibits the same behavior as in the case of quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. In addition, I present applications of the extension graph to the study of convergence actions, graph wreath products, and group von Neumann algebras when the defining graph is hyperbolic.

June 13th (Fri)

09:30 - 10:30Hyungryul Baik (KAIST)Title: Reconstruction of 3-manifolds from infinityAbstract: By work of Thurston, Calegari, Calegari-Dunfield, Fenley, and Frankel-Schleimer-Segerman, it is known that many 3-manifolds naturally have a circle at infinity which is usually referred as universal circle. We show that in many cases, we can reconstruct 3-manifolds from a universal circle. This talk is based on joint work with various subsets of KyeongRo Kim, Hongtaek Jung, Chenxi Wu, Bojun Zhao.

11:00 - 12:00Christian Rosendal (University of Maryland - College Park)Title: Aspects of automatic continuity 2Abstract: A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and refinements of the closed graph theorem, applying these to the problem of continuity of universally measurable homomorphisms and also to gauge the amount of choice needed to construct discontinuous homomorphisms between Polish groups.

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