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2024 YGTK 

(Meeting with Young Geometric Topologists at KAIST) 

Date: 2024.6.25 

Location: KAST E6-1, room 1401 

This is a one-day workshop with young geometric topologists. 

Schedule and abstracts 

§   10:00-11:00 Dongryul M. Kim (Yale University) 

Title: Conformal measure rigidity and ergodicity of horospherical foliations 

Abstract: As generalizations of Mostow's rigidity theorem, Sullivan, Tukia, Yue, and Oh and I proved rigidity theorems for representations of rank one discrete subgroups of divergence type, in terms of the push-forwards of conformal measures via boundary maps. In this talk, I will present a higher rank extension of them for a certain class of discrete subgroups, which we call hypertransverse subgroups. This class includes all rank one discrete subgroups, Anosov subgroups, relatively Anosov subgroups, and notably, their subgroups. The proof is developing the idea of the joint work with Oh for self-joinings of higher rank hypertransverse subgroups, overcoming the lack of CAT(-1) geometry on symmetric spaces. In contrast to the work of Sullivan, Tukia, and Yue, the argument is closely related to studying the ergodicity of horospherical foliations.

§   11:30-12:30 Sanghoon Kwak (KIAS)

Title: Big Out(F ₙ) and its rigidity  

Abstract: The group Out(F ₙ) 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(F ₙ)” as the mapping class group of (locally finite) infinite graphs. As a group of symmetries of graphs, one can ask how much of the group determines the graph, addressing the so-called rigidity question. In this talk I will present two types of rigidity questions for Big Out(F ₙ), and share partial progress toward answering each. This is a joint work with George Domat and Hannah Hoganson.

§   14:00-15:00 Philippe Tranchida (MPIM)

Title: The geometry of involutions in PGL(2,q) 

Abstract: I will first introduce the concept of coset geometries. This is a construction introduced by Jacques Tits, that allows to reconstruct a space a group is acting in from the group itself. The philosophy here is that all spaces a group can act "nicely" on should be visible in the algebraic structure of the group. I will then show how to use this concept to construct abstract polytopes and hypertopes (a generalization of polytopes) from the groups PGL(2,q), the groups of projectivities of a projective line over a finite field over order q. In order to do so, one has to get a good understanding of the involutions of PGL(2,q), which is best accomplished by seeing these involutions as perspectivities of a projective plane. If time permits, I will discuss potential generalizations to projective linear groups over arbitrary fields (or even division algebras). 

§   15:30-16:30 InHyeok Choi (KIAS/Cornell Univesity)

Title: Sullivan’s conjecture, Myrberg limit set and Hausdorff dimension

Abstract: In the 70’s, Patterson and Sullivan constructed measures at infinity, called the (quasi-)conformal measures, for Fuchsian and Kleinian groups that canonically ``sees” some geometry of the group action. Moreover, (quasi-)conformal measures are intimately related to the limit set of the groups and the dynamics of the geodesic flow on the quotient, with an application to the orbit counting problem. While studying this, Sullivan made a conjecture regarding typical geodesic ray in the ambient hyperbolic space. In this talk, I will survey the notions in this story and explain a recent result of Qing and Yang about sublinearly Morse directions. If time allows, I will explain an ongoing work regarding Hausdorff dimension of the Myrberg limit set.

§   16:45-17:45 KyeongRo Kim (SNU) 

Title: Uniform difference between the length spectra of Out(F2) and the genus two handlebody group

Abstract: Outer automorphism groups of free groups have been studied as an algebraic generalization of mapping class groups of surfaces of finite type. Especially, Bestvina, Hendel, Feighn, and many researchers studied irreducible elements in the outer automorphism group as an algebraic counterpart of pseudo-Anosov mapping classes. In this talk, we discuss handlebody groups as a bridge between outer automorphism groups of free groups and mapping class groups of finite type surfaces. In particular, I introduce a Hensel’s question about a connection between fully irreducible outer automorphisms and pseudo-Anosov mapping classes in terms of translation length. Also, I present a recent result about the genus-two case of the question. This is based on the joint work with Donggyun Seo.