The 10th KAIST Geometric Topology Fair

Past Events ( - 2011) Main Program Registration and Travel

August 13 – 17, 2012 the 10th KAIST GeometricRoom Topology1501, Natural Sciences Fair Building KAIST, Daejeon, Korea August 13 - 17, 2012. KAIST, Daejeon, Korea. http://mathsci.kaist.ac.kr/~shkim/gtfair

Lecture Series by and Research Talks by Jinseok Cho (KIAS) Michael Davis (Ohio State Univ.) Graph products, RACGs and RAAGs Kanghyun Choi (KAIST) Stefan Friedl (Univ. of Cologne) Koji Fujiwara (Tohoku Univ.) Thilo Kuessner (KIAS) Group actions on quasi-trees Sang-hyun Kim (KAIST) Taehee Kim (Konkuk Univ.) Alan Reid (Univ. of Texas at Austin) Sang-Jin Lee (Konkuk Univ.) 3-manifold groups, covering spaces Gye-Seon Lee (Seoul National Univ.) and LERF Seonhee Lim (Seoul National Univ.) Ken’ichi Ohshika (Osaka University)

The 10th KAIST Geometric Topology Fair will focus on the recent developments in geometric group theory and three-manifold theory. The lecture series and talks will Organizing committee: Suhyoung Choi, Sang-hyun Kim, Kihyoung Ko (KAIST) http://mathsci.kaist.ac.kr/be aimed at graduate students∼ shkim/gtfairand early career researchers. email:[email protected](National Research Fund of Korean Government, 2011-0026138 & 2011-0027600)

Organizing committee Kihyoung Ko, Suhyoung Choi, Sang-hyun Kim Contact [email protected] The 10th KAIST Geometric Topology Fair Page 1 of 7

Timetable

8/13 MONDAY 8/14 TUESDAY 8/15 WEDNESDAY 8/16 THURSDAY

9 - 9:30 Coffee and light breakfast

9:30 - 10:30 M. Davis M. Davis A. Reid A. Reid

11 - 12 K. Fujiwara K. Fujiwara K. Fujiwara M. Davis

12 - 1:30 Lunch

1:30 - 2:30 A. Reid S. Lee J. Cho S. Kim

2:30 - 3 Break

3 - 4 K. Ohshika T. Kuessner G. Lee S. Friedl

4:15 - 5:15 S. Lim T. Kim K. Choi

Reception

Lecture Series

Graph products, RACGs and RAAGs by Michael Davis (Ohio State Univ.) Given a graph W and a group G, one can define the graph product; it is the group K (=K(G,W)) formed by taking the free product of copies of G, one for each vertex of W, and then imposing the relations that copies of G corresponding to two distinct vertices commute whenever the vertices span an edge of K. When G is cyclic of order 2, K is a right-angled Coxeter group (abbreviated as RACG). When G is infinite cyclic, K is a right-angled Artin group (abbreviated as RAAG). I will explain two constructions of contractible spaces on which a graph product acts. One of these, called the right-angled building (= RAB) for K, is nonpositively curved, i.e., it is a CAT(0)-space. When K a RAAG, the second construction is also CAT(0). Next, I plan to discuss several applications of RACGs and RAAGs to topology and geometric group theory, for example, 1) the reflection group trick using RACGs and 2) the construction of Bestvina-Brady groups using RAAGs. I will also use the explicit description of these contractible spaces to compute the (co)homology of graph products with various coefficients. Group actions on quasi-trees by Koji Fujiwara (Tohoku University) The study on group actions on simplicial trees is classical. Bass-Serre theory gives a description of the group if the action is non-trivial, namely, there is no global The 10th KAIST Geometric Topology Fair Page 2 of 7

fixed point. It also has some connection to property T. A quasi-tree is a graph which is quasi-isometric to a simplicial tree. We will study group actions on quasi-trees by automorphisms. In this case, an action is non-trivial if there is an unbounded orbit. We give sufficient conditions for a group to have a non-trivial action on a quasi-tree, discuss standard examples, and study some consequence from non-trivial group actions on quasi-trees.

3-manifold groups, covering spaces and LERF by Alan Reid (Univ. of Texas at Austin)

Since Perelman’s resolution of the Geometrization Conjecture, the main open questions surrounding the geometry and topology of compact 3-manifolds re- volve around so called "virtual properties": i.e. on the geometry and topology of finite sheeted covers of compact 3-manifolds. Recent work of Agol now appears to have resolved many of these questions. These lectures will discuss work prior to Agol and will deal with residual properties of 3-manifold groups and finite sheeted covering spaces.

Research Talks

Introduction to Kashaev volume conjecture by Jinseok Cho (KIAS)

We explain the definition of Kashaev invariant of knots using R-matrix and in- troduce the volume conjecture. Especially, we follow Kashaev’s original idea of calculating the limit of the Kashaev invariant, and formulate his method as the optimistic limit of the Kashaev invariant. Then we revise his method and show that it determines the volume and the Chern-Simons invariant of the hyperbolic knot. This is a joint-work with Hyuk Kim and Seonhwa Kim of Seoul National University.

The definability criterion for cocompact convex projective polyhedral reflection groups by Kanghyun Choi (KAIST)

In this talk, we introduce the criterion for a Zariski dense subgroup generated by reflections Γ ⊂ SL±(n + 1, R) to be definable over A where A is an integrally closed Noetherian ring in a field R. We apply this criterion for groups gen- erated by reflections that act cocompactly on irreducible properly convex open subdomains of the n-dimensional projective sphere. This gives a methodology to construct injective group homomorphisms from such Coxeter groups to SL±(n + 1, Z). Finally we provide some examples of SL±(n + 1, Z)-representations of such Coxeter groups. In particular, we consider simplicial reflection groups that are isomorphic to hyperbolic simplicial groups and classify all the conjugacy classes of the reflection subgroups in SL±(n + 1, R) that are definable over Z. The 10th KAIST Geometric Topology Fair Page 3 of 7

Minimal genus surfaces in 4-manifolds with a free circle action by Stefan Friedl (University of Cologne)

We use recent results of Agol and Wise to give new lower bounds on the genera of surfaces representing homology classes in 4-manifolds with a free circle action.

Hyperbolic aspects of right-angled Artin groups by Sang-hyun Kim (KAIST)

For each right-angled Artin group G, we canonically associate a quasi-tree T. In the case when G has cohomological dimension two, this graph T precisely encodes all the isomorphism types of right-angled Artin groups that are embedded in G. In general, T provides a necessary condition for such isomorphism types. T turns out to be quasi-isometric to the coned-off Cayley graph of G relative to the centralizers of the vertices. We describe hyperbolic aspects of the action of G on this quasi-tree.

Knot concordance and invariants from derived covers by Taehee Kim (Konkuk Univ.)

Two knots are said to be concordant if there exists a properly locally flat embed- ded annulus in the product of the 3-sphere and the unit interval which gives the knots on the boundary.. In this talk, we discuss relationships between concor- dance and invariants from derived covers including the Alexander polynomial, higher-order Alexander invariants and von Neumann rho-invariants associated to rationally universal solvable representations defined by Cochran, Orr and Te- ichner. This is joint work with Se-Goo Kim.

Invariants preserved by mutation by Thilo Kuessner (KIAS)

Ruberman proved in 1987 that hyperbolic volume is preserved under mutation of knots. While that proof was building on results from the theory of minimal surfaces, we will show that Rubermans Theorem can also be derived from an equality in group homology, which has a short topological proof. This topologi- cal approach can also be used to show that several other invariants are preserved under (generalized) mutation of hyperbolic 3-manifolds, such as for example the volume of CR-structures or the volume of flag structures as considered by Bergeron-Falbel-Guilloux. If time permits we will also discuss generalizations to higher-dimensional symmetric spaces.

Real projective deformations of hyperbolic reflection orbifolds by Gye-Seon Lee (Seoul Na- tional Univ.)

Let Q be a compact reflection orbifold of dimension at least three. By using Klein’s model for hyperbolic geometry, hyperbolic structures on orbifolds provide exam- ples of real projective structures. If Q admits a hyperbolic structure, then such a hyperbolic structure is unique. However, the induced real projective structure on The 10th KAIST Geometric Topology Fair Page 4 of 7

some such orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. We will describe a large class of compact hyperbolic reflection orbifolds with such deformations. This is joint work with Suhyoung Choi.

Braid group of type (de,e,r) by Sang-Jin Lee (Konkuk Univ.)

We describe a Garside structure for the braid group of type (de,e,r). This gives an interpretation of the group as a subgroup of the Artin braid group. Using this, we show that the braid group of type (de,e,r) is strongly translation discrete and that roots are unique up to conjugacy. This is a jointwork with Ruth Corran and Eon-Kyung Lee.

Ford circles and Farey maps for function field by Seonhee Lim (Seoul National Univ.)

Gauss map (or continued fraction map) and Farey map are well investigated for real numbers. It is related to geodesic flow on the hyperbolic plane. We define analogous objects on the Serre’s tree of SL2 over function field and show that there is a finite measure invariant under Farey map.

Actions of isomorphism groups of Heegaard splittings on projective lamination spaces by Kenichi Ohshika (Osaka Univ.)

Sakuma and D. Lee showed interesting properties of simple closed curves in 2-bridge link complements using actions of isomorphism groups on the Farey graph. We shall show that we can generalise some of their results in the set- ting of general Heegaard splittings or bridge decompositions using bi-Lipschitz models of Heegaard splittings develeped by H. Namazi. In particular, we shall show that the action of isomoprhism group on the projective lamination space has non-empty domain of discontinuity. This is a part of joint work with Makoto Sakuma. The 10th KAIST Geometric Topology Fair Page 5 of 7

Hotels: Daedeok Innopolis Guesthouse or Toyokyo Inn

From/to KAIST : Take a taxi. Or, it is a good 3 km (Guesthouse) or 4.5 km (Inn) walk. Airport Limousine Bus : The bus journey takes 2 hours and 40 minutes from Incheon International Airport to Daejeon. Get off at Daejeon Government Complex Stop or Lotte Hotel Stop, and then take a taxi. Train : From Daejeon train station, it takes 25 minutes by taxi. Express Bus : From the government complex bus stop or Yuseong bus terminal, take a taxi. Daejeon ←→ Seoul (Kyeongbu Exp. Way/153.2 km) : 2 hours The 10th KAIST Geometric Topology Fair Page 6 of 7

KAIST campus map / facilities

Medical Center - E21 ··························· Central Library/Bookstore - E9 East Cafeteria - E5 1F(students) / 2F(faculties) ········· North Cafeteria - N11 1F West Cafeteria - W2 1F ································· Gran-K restaurant - N13 2F Il Mare: Grazioso (Italian Restuarant) - W2 2F ·················· DDDN Pizza - W2 1F Burger King - N12 1F ······························ Dunkin’ Donuts - E3 1F Caffe Bene - E4 1F/N11 ······························ Coffee Bean - W2-1 Convenience Stores - N12,W4,W3,E5,E8

If you need more information, visit: http://www.kaist.edu/english/01_about/map.htm (campus map / buildings) http://www.kaist.edu/english/04_campus/03_welfare_04.php?pt=4 (shops, restaurants and more in KAIST) The 10th KAIST Geometric Topology Fair Page 7 of 7

Participants

* speakers Byunghee An, POSTECH, Pohang Jinseok Cho*, KIAS, Seoul Kanghyun Choi*, KAIST, Daejeon Michael Davis*, Ohio State University, Columbus Stefan Friedl*, University of Cologne, Cologne Koji Fujiwara*, Tohoku University, Sendai S. Ghorbanifar, Tarbiat Modares University, Tehran Jiyoung Han, Seoul National University, Seoul SangEon Han, Chonbuk National University, Jeonju Seung Uk Jang, KAIST, Daejeon Seong Gu Jeong, KAIST, Daejeon Gyo Taek Jin, KAIST, Daejeon Yewon Joung, Pusan National University, Busan Sungkyung Kang, KAIST, Daejeon Jee Hyoun Kim, Konkuk University, Seoul Jisu Kim, Seoul National University, Seoul Jongtae Kim, KAIST, Daejeon Sang-hyun Kim*, KAIST, Daejeon Seonhwa Kim, Seoul National University, Seoul Sujin Kim, Seoul National University, Seoul Taehee Kim*, Konkuk University, Seoul Youngju Kim, KIAS, Seoul Thilo Kuessner*, KIAS, Seoul Sanghoon Kwon, Seoul Nationl University, Seoul Dongsoo Lee, Pusan National University, Busan Donsung Lee, KAIST, Daejeon Eon-Kyung Lee, Sejong University, Seoul Gye-Seon Lee*, Seoul National University, Seoul Sang-Jin Lee*, Sejong University, Seoul Sunhyuk Lim*, KAIST, Daejeon Kenichi Ohshika*, Osaka University, Osaka HyoWon Park, KAIST, Daejeon Alan Reid*, University of Texas at Austin, Austin Khosro Tajbakhsh, Tarbiat Modares University, Tehran

Leaflet made by Donsung Lee