# Geometric Topology Fair in Korea (2016)

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This page can be accessed at http://gtfair.cayley.kr

Geometric Topology Fair in Korea (2016) is an annual conference series encompassing geometric topology, geometric group theory and dynamics. It focuses on interactions between geometric group theory and low-dimensional topology. The mini courses are particularly aimed at graduate students and early career researchers.

## Program

### Schedule

#### Special slots

• July 4 (M) 5 pm: Bestvina KAIST CMC Distinguished Lecture (reception follows at Faculty Club).

#### Time Table

 July 5 (Tu) July 6 (W) July 7 (Th) July 8 (F) 9 - 9:30 Coffee 9:30 - 10:30 Bestvina I Fujiwara I Bestvina II Fujiwara II 10:40 - 11:40 Eskin I Rafi I Eskin II Rafi II 1:30 - 2:30 Navas I Navas II Lim 2:40 - 3:40 Han Kim Kwon 3:50 - 4:50 O Kuessner

### Abstracts

KAIST CMC Distinguished Lecture (July 4, 5 pm)

Mladen Bestvina, On the large-scale geometry of mapping class groups

As envisioned by Gromov, geometric group theory is the study of large-scale geometry of groups. The key idea is to study groups as metric spaces. Examples of large-scale invariants are the isoperimetric function and asymptotic dimension, and I will focus on the latter. The class of groups where this is all well understood is the class of hyperbolic groups. However, groups of interest are often not hyperbolic, but they can be sometimes understood in terms of hyperbolicity. The basic example is the mapping class group of a surface. If time permits, I will explain the main ideas in the proof, due to Bromberg, Fujiwara and myself, that mapping class groups have finite asymptotic dimension.

#### Minicourses

Two talks per each.

Mladen Bestvina, On the Farrell-Jones conjecture for mapping class groups

I will try to describe what the Farrell-Jones conjecture is about, and how one goes about proving it. Then I will try to outline a proof of FJC for mapping class groups, which is work in progress, joint with Arthur Bartels.

Alex Eskin, Polygonal Billiards and Dynamics on Moduli Spaces

Billiards in polygons can exhibit some bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry (and in particular Hodge theory), Teichmuller theory and ergodic theory on homogeneous spaces. I will attempt to give a gentle introduction to the subject, assuming virtually no background. Most of the first talk will be accessible to undergraduates and first year graduate students.

Koji Fujiwara, Acylindrically hyperbolic groups are not invariably generated

We discuss acylindrical actions on hyperbolic spaces and some application. A group G is "invariably generated" if there is no proper subgroup H in G such that any element in G is conjugate to some element in H. We prove that an acylindrically hyperbolic group is NOT invariably generated. This is a joint work with Bestvina.

Andrés Navas, Group actions on the circle in low regularity: rigidity and flexibility

In these two talks we will show several examples of constraints on the regularity of group actions on 1-dimensional spaces. We will discuss the case of Baumslag-Solitar groups, nilpotent groups, groups of intermediate growth, etc, mostly focusing on very low regularity.

Kasra Rafi, Balls in Teichmüller space are not convex. Preliminary report.

We prove that when 3g - 3 + p > 1, the Teichmüller space of the closed surface of genus g with p punctures contains balls which are not convex in the Teichmüller metric. We analyze the quadratic differential associated to a Teichmüller geodesic and, as a key step, show that the extremal length of a curve (as a function of time) can have a local maximum. This is a joint work with Maxime Fortier Bourque.

#### Research Talks

Jiyoung Han (Seoul National University), On a quantitative S-arithmetic Oppenheim conjecture

We prove a generalization of a theorem of Eskin-Margulis-Mozes in an S-arithmetic setup: suppose that we are given a finite set of places S over ℚ containing the archimedean place, an irrational isotropic form q of rank n≥4 on ℚ_S, a product of p-adic intervals I_p, and a product Ω of star-shaped sets. We show that if the real part of q is not of signature (2,1) or (2,2), then the number of S-integral vectors v in 𝖳Ω satisfying simultaneously that q(v) is contained in I_p for p∈S is asymptotically proportional to T^{n-2}, as T goes to infinity, where the proportional constant depends only on the quadratic form, Ω and the product of Haar measures of I_p, p∈S. This is a joint work with Seonhee Lim and Keivan Mallahi-Karai.

Sang-hyun Kim (Seoul National University), Obstruction for a virtual C^2 action on the circle.

When does a group virtually admit a faithful C^2 action on the circle? We provide an obstruction using a RAAG. Examples include all (non-virtually-free) mapping class groups, Out(Fn) and Torelli groups. This answers a question by Farb, and is analogous to Ghys and Burger--Monod characterization of higher rank lattice actions on the circle. (Joint work with Hyungryul Baik and Thomas Koberda)

Thilo Kuessner (KIAS), On measure homology of some "wild" spaces.

Measure homology is a variant of singular homology which uses measures on the space of simplices rather than just finite sums of simplices. It was introduced by Thurston to effectively compute the simplicial volume of hyperbolic manifolds. There is a canonical map from singular homology to measure homology and it is known to be an isomorphism for all CW-complexes. However it need not be an isomorphism for "wild" (i.e., not semi-locally simply connected) spaces. Classical examples of such "wild" spaces are shrinking wedges, that is, the one-point union of a countable sequence of spaces whose diameter tends to zero. We consider shrinking wedges of negatively curved manifolds and show that the canonical map from singular to measure homology is injective in this case. (This is joint work with Janusz Przewocki and Andreas Zastrow.)

Sang-hoon Kwon (Seoul National University), On spectral gaps on trees

Exponential mixing property of the geodesic flow in negatively curved spaces is comprehensively exhibited, especially for finite volume CAT(-1) spaces and geometrically finite hyperbolic manifolds. We consider non-Archimedean analogue for infinite volume cases, namely the geodesic flow on the quotient of trees. We present the sufficient condition concerning spectral gap about the group for which the geodesic translation map on the quotient is mixing with exponential rate. These examples include Schottky free groups and full groups associated to geometrically finite discrete groups acting on trees.

Seonhee Lim (Seoul National University), On Sturmian colorings on trees

We will define subword complexity of vertex colorings of trees and give various examples including examples related to commensurators of tree lattices. We will show an induction algorithm for Sturmian colorings which are colorings of minimal unbounded subword complexity. This is a joint work with Dong Han Kim.

Sangrok O (KAIST), Quasi-isometric classification of planar graph 2-braid groups

Let $$G$$ be a planar graph. We classify the quasi-isometric types of planar graph 2-braid groups, $$B_2(G)$$. In fact, we use an axillary finite complex $$M(G)$$ constructed from a given planar graph $$G$$. If there exists a component of $$M(G)$$ such that it has an induced rotation symmetry, but does not have a reflection symmetry along cut vertices, then $$B_2(G)$$ is not quasi-isometric to a right-angled Artin group(RAAG) but instead quasi-isometric to a finitely iterated HNN extension of a RAAG whose defining graph is bipartite. Otherwise, $$B_2(G)$$ is quasi-isometric to a RAAG whose defining graph is bipartite.

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## Organizing committee

Sang-Jin Lee (Konkuk University)
Sang-hyun Kim (Seoul National University)
Seonhee Lim (Seoul National University)
Kihyoung Ko (KAIST)

## Short-term visitors at KAIST (2016)

Mladen Bestvina
• short-term course at KAIST on Tues/Wednesdays, 11 am / 2 pm for June 21 - July 13.
• lectures on June 21 - 22, 28 - 29, July 12 - 13.
Koji Fujiwara
• June 27 (M) – July 9 (Sa)
Kasra Rafi
• July 4 (M) – July 16 (Sa)