# Past seminars

## 2021

**March 15, 17 (M, W), 2 - 3:30 pm, Zoom**

Junehyuk Jung (Brown)

*Equidistribution problems of closed geodesics on hyperbolic surfaces*

In this series of lectures, I will introduce the modular intersection kernel and then explain how one can use this to understand intersections of geodesics using it.

(1) In the first lecture I'm going to go over some background topics that are necessary for understanding the main result. This includes Duke's theorem on equidistribution of closed geodesics on the full modular surface $\mathbb{X}=PSL_2(\mathbb{Z})\backslash \mathbb{H}$, discriminant geodesics, period integrals, spectral decomposition of $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$, Selberg's trace formula, the prime geodesic theorem, etc. This is a comprehensive lecture designated for non-experts, so those who are already familiar with these concepts may skip it.

(2) Two theorems regarding the intersections of closed geodesics on the full modular surface will be introduced in the second lecture. Let $C_d$ be the union of closed primitive geodesics corresponding to indefinite primitive binary quadratic form of discriminant $d$. Then a generalized form of Duke's theorem implies that the lift $\mathscr{C}_d$ of $C_d$ to the unit tangent bundle $S\mathbb{X}$ of $\mathbb{X}$ becomes equidistributed as $d\to \infty$. As an application of generalized Duke's theorem to the modular intersection kernel, for any fixed geodesic segment $\beta$, the intersections between $\beta$ and $C_d$ become equidistributed on $\beta$ as $d \to \infty$. This resolves the main conjecture introduced by Rickards based on numerical experiment. The second theorem I will talk about provides an asymptotic formula on the total number of intersections between $C_{d_1}$ and $C_{d_2}$.

This lecture is based on the collaboration with Naser Talebizadeh Sardari.

## 2020

**June 9, 2020 (Tu), 4 pm**

Xinghua Gao (KIAS)

*The L-Space Conjecture of Closed 3-Manifolds*

Boyer, Gordon, and Watson conjectured that for an irreducible rational homology 3-sphere, not being an L-space, its fundamental group being left-orderable and admitting a taut foliation are equivalent. This conjecture, known as the L-space conjecture, relates many aspects of 3-manifolds together, such as lamination, foliation, Heegaard Floer homology, and group action on the real line. In this talk, I’m going to talk about my work related to the conjecture, especially the left-orderability side of the conjecture.

**June 23, 2020 (Tu), 4 pm**

Jongbaek Song (KIAS)

Introduction to Polyhedral Products and Its Application to Geometric Group Theory.

A polyhedral product is a subspace of a Cartesian product that is parametrized by a simplicial complex together with a pair of spaces. Depending on the choice of ingredients, it has a wide range of applications in many different areas of mathematics such as toric geometry, homotopy theory, topological data analysis, geometric group theory, and so on. In this *survey* talk, we briefly overview the definition and properties of a polyhedral product, and study how it is applied to right angled Artin/Coxeter groups and relevant objects in geometric group theory.

**July 7, 2020, 4 pm**

Donggyun Seo (KIAS)

*RAAGs generated by powers of Dehn twists*

It is well-known that mapping class groups of negatively curved surfaces contain infinitely many conjugate classes of right-angled Artin subgroups (abbrev. RAAGs). By Koberda, if Dehn twists and pseudo-Anosov’s on essential subsurfaces satisfy some mild condition, their sufficiently large uniform powers generate a RAAG. In this talk, I will focus on how large powers of Dehn twists are required to generate a RAAG, related with the geometric intersection number.

**August 4, 2020 (Tu), 4 pm**

Inkang Kim (KIAS)

*New Kahler metrics on Teichmuller space and harmonic maps *

We survey several known facts on Teichmuller space and present a new way of constructing Kahler metrics on Teichmuller space via harmonic map techniques. This is a joint work with Wan and Zhang.

**September 1, 2020 (Tu), 9 - 10 am**

Homin Lee (Indiana University)

*Hyperbolic dynamics in the rigidity of group actions*

We will survey the use of hyperbolic dynamics in the rigidity of group actions on manifolds. We will focus on higher rank free abelian group or higher rank lattice actions that have some uniform hyperbolicity such as Anosov diffeomorphism or flow. We will discuss old and new results with some ideas of the proof if time permits.

**September 8, 2020 (Tu), 4 pm**

Seonwoo Kim (Seoul National University)

*Second time scale of the metastability of reversible inclusion processes*

We briefly review the timeline of research on the inclusion process and metastability on the model. Then, we investigate the second time scale of the metastable behavior of the reversible inclusion process in an extension of the study by [Bianchi, Dommers, and Giardinà, Electronic Journal of Probability, 22: 1-34, 2017], which presented the first time scale of the same model and conjectured the scheme of multiple time scales.

**September 22, 2020 (Tu), 4 pm**

Jihoon Park (Korea University)

*Local topology of the space of kleinian punctured torus group*

The deformation space AH(M) of hyperbolic 3-manifold is the space of all hyperbolic 3-manifold with fixed homotopy type. The interior of AH(M) is well-understood since 1970, but the topology of the entire space is more complicated by existence of singular points in the boundary. In this talk, we give almost complete classification of local singularity in the simplest case of deformation space, the space of once punctured torus group.

**October 1****3**** (****Tu****) ****9**** pm, Korea = 8 am ****Eastern Standard Time, US**** (note the time!)**

Sam Nariman (Purdue)

Mather-Thurston’s theory and group homology of diffeomorphism groups

In this mostly expository talk, I will discuss a remarkable generalization of Mather’s theorem by Thurston that relates the identity component of diffeomorphism groups to the classifying space of Haefliger structures. The homotopy type of this classifying space plays a fundamental role in foliation theory. However, it is notoriously difficult to determine its homotopy groups. Mather-Thurston theory relates the homology of diffeomorphism groups to these homotopy groups. Hence, this h-principle type theorem has been used as the main tool to get at the homotopy groups of Haefliger spaces.

There are different approaches to Mather-Thurston’s theory. There exist lecture notes by Mather of Thurston’s talk on this theorem which is our main source of inspiration. McDuff and Segal developed the homotopy theory of topological monoids and gave a proof using monoids of embeddings. McDuff used their method to prove Mather-Thurston theory also for volume preserving diffeomorphisms. Gael Meigniez recently found a geometric proof of this theorem using ``local models” to fill the holes by foliations and Mike Freedman also in low homological degrees improved this theorem for C^0-foliations by ``global models” to fill the holes. However, we mainly discuss the original idea of Thurston which I think is remarkable mainly because unlike other h-principle theorems and also the method McDuff-Segal, he does not use a ``local statement” to get to a ``global statement”. I will mention how his method can be improved to give Mather-Thurston type theorems for other transverse structures.

## 2019

**August 16 (F), 4 - 5:30, Room 1424**

Koberda, Thomas (University of Virginia)

*Commensurations of thin groups*

I will discuss a recent result joint with M. Mj, which shows that the commensurator of an infinite index normal subgroup of an arithmetic lattice in a simple Lie group is discrete in most cases.

**August 21 (W), 9 - 5:30, Room 8101**

KIAS workshop on low-dimensional topology

http://events.kias.re.kr/h/WLDT19/

**August 22 (Th), 4 - 5:30, Room 1423**

Ohshika, Kenichi (Gakushuin University)

*The rigidity of mapping class group actions*

The mapping class group of a closed surface S acts naturally on various spaces related to S preserving their structures : for instance, the Teichmuller space of S with the Teichmuller metric and the Weil-Petersson metric, the curve complex of S with its combinatorial structure, the measured lamination space of S with the intersection form, the geodesic lamination space of S with the Hausdorff topology etc.

In this talk, I will explain that such natural actions constitute the total group of automorphisms for four of them: the measured lamination space with the intersection form, the unmeasured lamination space with the quotient topology, the reduced Bers boundary of Teichmuller space, and the geodesic lamination space with the left Hausdorff topology.

This is partly joint work with A. Papadopoulos.

**August 27 & 29 (T & Th), 2 - 3:30, Room 1423**

Michele Triestino (Univ. Bourgogne Franche-Comté)

*On groups of real analytic circle diffeomorphisms I*

In these two talks we give an overview on the study of finitely generated groups of real analytic circle diffeomorphisms, which has a long history, motivated by questions from foliation theory and influenced by the more modern geometric group theory.

In the first part, we will discuss non discrete subgroups, showing that their dynamics is very rich and almost continuous.

In the second part, we will discuss discrete subgroups, which constitute a topic of more current study. As sample result, if the action is not minimal, then it is a ping pong action of a virtually free group.

**September 24 (T), 4 pm, Room 1423**

Tan, Ser Peow (National Univ. of Singapore)

*A probabilistic and combinatorial approach to McShane’s identity*

We give a probabilistic interpretation of McShane’s identity in terms of the probability of going along certain infinite paths along some combinatorial tree. This follows ideas of Bowditch in the case of the once punctured torus and is joint work with Francois Labourie.

**October 2 (W), 2 pm, Room 1423 (note the time!)**

Jung, Junehyuk (Texas A&M University)

Classification of embedded closed totally geodesic surfaces in Bianchi 3-folds.

In a recent work with Alan Reid, we showed that the Bianchi 3-fold $\Gamma_d\backslash \mathbb{H}^3$ has at least one embedded closed totally geodesic surface if and only if $d$ is not one of

1,~2,~3,~5,~6,~7,~10,~11,~15,~19,~21,~35, 51

The proof uses Polya-Vinogadov inequality, Siegel-Watson theorem, and some theory of binary Hermitian forms over an imaginary quadratic field. In this talk, I will go over the proof and illustrate a few applications. (This is a joint work with Alan Reid.)

**October 21 (M), 4 pm, Room 1424 (note the place!)**

Andrei Yu. Vesnin (Tomsk State University and Sobolev Institute of Mathematics, Russia)

*Right-angled hyperbolic Coxeter groups, census of polyhedra and construction of 3-manifolds*

A study of existence of bounded right-angled polyhedra in hyperbolic 3-space with started by Pogorelov [1]. These polyhedral are referred as to Pogorelov polyhedra now. The methods to construct hyperbolic 3-manifolds (orientable and non-orientable) related to corresponding right-angled Coxeter groups is described in [2]. Moreover, Pogorelov polyhedra play an important role in the toric topology due to cohomological rigidity of manifolds related to them [3]. The census of bounded right-angled hyperbolic polyhedral is presented in [4].

We will present a survey of results of Pogorelov polyhedral and construction of 3-manifolds from them. Also, we will present some results on census of finite-volume right-angled hyperbolic polyhedral with all vertices on the absolute of hyperbolic 3-space [5].

References:

[1] A. Pogorelov, A regular partition of Lobachevskian space, Math. Notes, 1:1 (1967), 3–5.

[2] A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds, Russian Math. Surveys, 72:2 (2017), 335–374.

[3] V. Buchstaber, N. Erokhovets, M. Masuda, T. Panov, S. Park, Cohomological rigidity of manifolds defined by 3-dimensional polytopes, Russian Math. Surveys, 72:2 (2017), 199–256.

[4] T. Inoue, Exploring the list of smallest right-angled hyperbolic polyhedra, Experimental Mathematics, Published online April 11, 2019.

[5] A. Vesnin, A. Egorov, Ideal right-angled polyhedra in Lobachevsky space, 20 p. Preprint arxiv:1909.11523, 2019.

**October 22 (T), 4 pm, Room 1423**

Song, Jongbaek (KIAS)

Manifolds with locally standard torus actions and their equivariant cohomology algebras.

We discuss symplectic toric manifolds and contact toric manifolds from the toric topological point of view. Those two classes of manifolds share certain similar properties, so-called “local standards”, which allows us to recover the original manifold from the orbit space together with torus action data. We study their equivariant cohomology algebras and observe how to use them for the equivariant classification of those manifolds.

**October 29 (T), 4 pm, Room 1423**

Cho, Cheol-Hyun (Seoul National University)

*Milnor fibers of curve singularities and mirror symmetry*

First, we survey some known results regarding Milnor fibers of polynomials of two variables. Then, we explain Homological mirror symmetry conjecture on invertible ones (known as Berglund-Hubsch conjecture), and report a work in progress which shows a relation to Auslander-Reiten quiver of Cohen-Maculay modules for ADE curve singularity. This is a joint work with Dongwook Choa and Wonbo Jung)

**November 19 (T), 4 pm, Room 1423**

Sakuma, Makoto (Hiroshima University)

*Kleinian groups generated by two parabolic transformations*

I will explain a complete proof to Agol’s classification of non-free Kleinian groups generated by two parabolic transformations, obtained by joint work with Hirotaka Akiyoshi, Ken’ichi Ohshika, John Parker, and Han Yoshida. I would also like to present a conjectural picture of the space of such Kleinian groups, which is a joint work in progress with Hirotaka Akiyoshi, Gaven Martin, Ken’ichi Ohshika, John Parker.

**December 10 (T), 4 pm, Room 1423**

Dami Lee (University of Washington)

*A geometric realization of Fermat's quartic*

In this talk, we will study a triply periodic polyhedral surface whose vertices correspond to the Weierstrass points on the underlying Riemann surface. The symmetries of the surface allow us to construct hyperbolic structures and various translation structures that are compatible with its conformal type. With this explicit data, one can find its algebraic description, automorphism group, Veech group, etc.

**November 24 (Tu)**** ****11**** ****a****m, Korea = ****Mon 9**** ****p****m ****EST, US ****(note the time!)**

Jingyin Huang (Ohio State Univ)

*Measure equivalence classification of certain right-angled Artin groups*

The notion of measure equivalence between countable groups was introduced by Gromov as a measure-theoretic analogue of quasi-isometry. In this talk we look at the problem of classifying right-angled Artin groups up to measure equivalence, and we will show that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. On the other hand, we will also mention some aspects of measure equivalence classification of right-angled Artin groups which are dramatically different from the quasi-isometric classification. This is joint work with Camille Horbez.

**January 7 (T), 4 pm, Room 1423**

Cristóbal Rivas (Universidad de Santiago de Chile)

*Regularity of orderable groups*

I will discuss about (left) orderable groups, which are groups that naturally act on the line by homeomorphisms. I will focus on the question on how smooth can a group or a group-action be made.

**January 9 (Th), 4 pm, Room 1423**

Hung C Tran (University of Oklahoma)

*Strong quasiconvexity and almost malnormality*

Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and it is preserved under quasi-isometry. It is well-known that quasiconvex subgroups of hyperbolic groups are finitely generated and have finite height (a generalization of almost malnormality). I prove that strongly quasiconvex subgroups of arbitrary finitely generated groups are also finitely generated and have finite height. Therefore, it may be interesting to ask if a finitely generated, finite height (or even almost malnormal) subgroup is strongly quasiconvex. I will talk about the answer to this question in $3$--manifold groups. A part of my talk is joint work with Hoang Thanh Nguyen and Wenyuan Yang.

**February 18 (Tu), 2 pm, Room 1423**

Homin Lee (Indiana U)

*Rigidity theorems for actions of higher rank lattices*

We will talk about rigidity theorems for smooth actions of a higher rank lattice on compact manifolds following the philosophy of the Zimmer program. Let be a semisimple Lie group. Assume that all simple factors of have a higher rank. If is a lattice of such a , then many rigidity phenomena are known due to the presence of "higher rank" and "property (T)". In this case, Zimmer's cocycle superrigidity theorem plays an important role. If higher rank lattice does not have the property (T), however, we can not expect Zimmer's cocycle superrigidity theorem to hold.

In this talk, we will consider higher rank lattices without property (T) such as . We will see local and global rigidity theorems for such a lattice action despite the absence of property (T). The main ingredient will be a dynamical superrigidity theorem that is an analog of Zimmer's cocycle superrigidity.

# 2016

## Keivan Mallahi-Karai (Jacobs University)

December 12th (Monday), 4:00 pm - 5:00 pm, Bldg. 129-301

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*On the chromatic number of structured Cayley graphs*

## Lim Seonhee (Seoul National University)

December 6th (Tuesday), 4:00 pm - 5:00 pm, Bldg. 129-406

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*Around quantitative Oppenheim conjecture*

## Lei Zhao (Chern Institute of Mathematics)

November 30 (Wednesday), 4:00 pm - 5:00 pm, Bldg. 129-406 (to be confirmed)

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*Relative equilibrium motions and real moment map geometry*

## Hyowon Park (Seoul National University)

November 22 (Tuesday), 4:00 pm - 5:00 pm, Bldg. 129-104

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*Finite index subgroups of right-angled Artin groups*

## Sang-hyun Kim (Seoul National University)

November 16 (Wednesday), 4:30 pm - 5:30 pm, Bldg. 27-116

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*Flexibility of projective representations*

## Paul Jung (KAIST) (Topology/dynamics joint seminar)

November 8 (Tuesday), 4 pm - 5 pm, Bldg. 129-104

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*An alpha-stable limit theorem for Sinai billiards with cusps*

## Yoshifumi Matsuda (Aoyama Gakuin University)

October 24 (Monday), 4 pm - 5 pm, Bldg. 27-325

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*Bounded Euler number of actions of 2-orbifold groups on the circle*

## JaeChoon Cha (Postech)

October 6 (Thursday), 4 pm - 5 pm, Bldg. 129-101

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*4-manifold topology and disk embedding (Colloquium)*

## BoGwang Jeon (Columbia University)

August 8 (Monday), 11 am - 12 pm, Bldg. 129-301

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*The Unlikely Intersection Theory and the Cosmetic Surgery Conjecture*

## Sejong Park (National University of Ireland Galway)

August 2 (Tuesday), 4 - 5 pm, Bldg. 129-301

[Expand]

*Toward a structure theorem for double Burnside algebras*

## Insuk Seo (UC Berkeley)

July 20 (Wednesday), 4 - 5 pm, Bldg. 129-301

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*Metastable behavior of the dynamics perturbed by a small random noise*

## Urs Frauenfelder (University of Augsburg)

July 15 (Friday), 11am - 12 pm, Bldg. 129-301

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*Periodic orbits in the restricted three body problem and Arnold's J^+ invariant.*

## Thomas Koberda (University of Virginia)

July 14 (Thursday), 4 - 5 pm, Bldg. 129-301

*Group actions on low dimensions I*

July 28 (Thursday), 4 - 5 pm, Bldg. 129-406

*Group actions on low dimensions II*

## Kasra Rafi (University of Toronto)

July 12 (Tuesday), 4 - 5 pm, Bldg. 129-301

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*The Teichmüller diameter of the thick part of moduli space*

## Jaesuk Park

May 20 (Friday), 2 - 4 pm, Bldg 25-103

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*Non-commutative quantum field theory and Chen’s iterated path integrals*

## Mounir Nisse

May 19 (Thursday), 4 - 6 pm, Bldg 25-103

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*Amoebas and Coamoebas*

May 20 (Friday), 4 - 6 pm, Bldg 25-103

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*On Coamoebas of Algebraic Hypersurfaces*

## Koji Fujiwara (Kyoto University)

May 11 (Wednesday), 5 - 6 pm, Bldg 129-104

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*Handlebody subgroups in mapping class groups*

May 12 (Thursday), 4 - 5 pm (colloquium), Bldg 129-101

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*Quasi-homomorphisms into non-commutative groups*

## Min Lee (University of Bristol)

May 4 (Wednesday) 2-3pm, 129-406

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*Effective equidistribution of primitive rational points on expanding horospheres*

## Jae Choon Cha (Postech)

Apr 27 (Wednesday), 5 - 6 pm, Bldg. 129-104

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*Quantitative topology and Cheeger-Gromov universal bounds*

## Richard M. Weiss (Tufts University)

Apr 5 (Tuesday), 5 - 6 pm, , Bldg. 129-104.

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*Simple groups and buildings*

## Masato Mimura (Tohoku University)

Mar 22 (Tuesday), 3:30 - 4:45 pm, Bldg. 24-207

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*Expanders and Margulis' construction*

Mar 23 (Wednesday), 2 - 3 pm, , Bldg. 129-406

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*Strong algebraization of fixed point properties*

Mar 24 (Thursday), 3:30 - 4:45 pm, Bldg. 24-207

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*Non-commutative universal lattices and unbounded rank expanders*

## Ilya Gekhtman (Yale Univ. and Univ. of Bonn)

March 18 (Friday) 2-3pm, 129-301. Dynamics of convex cocompact subgroups of mapping class groups

## Giulio Tiozzo (Yale Univ.)

March 16 (Wednesday) 2-3pm, 129-406. Random walks on weakly hyperbolic groups

Mar 17 (Thursday) 4-5pm, 129-101 auditorium (Colloquium talk)

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Random walks on weakly hyperbolic groups

## Junehyuk Jung (KAIST)

Mar 16 (Wednesday), 5 - 6 pm, Bldg. 129-104.

*Nodal domains of eigenfunctions on chaotic billiards*

[Expand]Abstract

## Hyungryul Baik (MPI Bonn)

Mar 3, 3:30 - 4:45, Room 24-207,

*Orderability*Mar 8, 3:30 - 4:45, Room 24-207,

*Laminarity*Mar 8, 5:00 - 6:00, Room 129-301,

*Convergence group*

[Expand]

Abstract

## 강정수

February 16 - 18, 4:30 - 6pm, Room 129-301.

*Symplectic capacities, Old and New I, II, III*

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Abstract

## Thomas Koberda (University of Virginia)

January 7, 4 pm - 6 pm, 129-301,

*Exotic quotients of surface groups*

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Abstract

## HyunKyu Kim (KIAS)

*Topological approach to seed-trivial mutation sequences in cluster algebras*January 4, 2 pm, 129-301.

# 2015

At SNU.

## November 25, Hyun-shik Shin (KAIST)

*Pseudo-Anosov mapping classes not arising from Penner's construction*

In this talk, we will discuss one property that is shared by all pseudo-Anosov mapping classes from Penner's construction. That is, all Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner’s construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner’s construction. This resolves a conjecture of Penner. This is a joint work with Balazs Strenner.

4 pm, 129-104.

## August 9–13, Geometric Topology Fair in Korea

### Minicourses

Fujiwara, Koji (Kyoto University), Contracting geodesics and acylindrical actions

Wilton, Henry (University of Cambridge), Profinite completions of 3-manifold groups

### Research Talks

Kim, Sungwoon (KIAS), Simplicial volume, Barycenter method and Bounded cohomology

Kuessner, Thilo (KIAS), Chern-Simons invariants of 3-manifold groups in SL(4,R)

Kwon, Sanghoon (Seoul National University), Effective Mixing and Counting in Trees

Ohshika, Ken'ichi (Osaka University), Accumulation of closed 3-manifolds within character varieties

## July 9, Seung-wook Jang (University of Chicago)

## January 16–21, Thomas Koberda (Yale)

Lecture I. The relationship between right-angled Artin groups and mapping class groups

In this general talk, we will give an overview of work joint with S. Kim concerning the relationship between right-angled Artin groups and mapping class groups. We will consider the broad question of, what right-angled Artin subgroups does a mapping class group or a right-angled Artin group admit?

Lecture II. The curve complex for a right-angled Artin group

In this talk we will give a more in depth discussion of the analogy between right-angled Artin groups and mapping class groups, through the comparison of the geometry of the extension graph and the curve complex.

Lecture III. Convex cocompactness for subgroups of right-angled Artin groups

In this talk we will discuss a new result joint with J. Mangahas and S. Taylor which characterizes finitely generated subgroups of right-angled Artin groups which have quasi-isometric orbit maps on the extension graph. These are analogous to convex cocompact subgroups of mapping class groups as defined by B. Farb and L. Mosher, and which are still rather poorly understood despite having attracted so much attention in recent years.

Jan 16 (2pm), 19 (4pm), 21 (4pm), Bldg 129-301

## January 13, BoGwang Jeon (Columbia)

*Hyperbolic three manifolds of bounded volume and trace field degree*

In this talk, I present my recent proof of the conjecture that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

4 pm, 129-301

# 2014

At SNU.

## December 9, Piotr Przytycki (McGill University)

*Balanced wall for random groups*

Gromov showed that one way to obtain a word-hyperbolic group is to choose a presentation "at random". I will survey random group properties in Gromov's model at various values of the density parameter. We will then focus on Ollivier-Wise cubulation of random groups for density parameter <1/5. I will indicate how to construct new walls that work at higher densities. This is joint work with John Mackay.

5 pm.

### Related classes at SNU

12/09, 12/10

*Systolic group I, II*

### Related talk at KAIST

2014-12-11 (15:00 - 16:00)

*Arcs intersecting at most once*

We prove that on a punctured oriented surface with Eulercharacteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.

## November 13, Alden Walker (University of Chicago)

*Random groups contain surface subgroups*

Gromov asked whether every one-ended hyperbolic group contains a surface subgroup. I'll explain this question and sketch the proof that a random group (an example of a one-ended hyperbolic group) contains a surface subgroup. I'll give all necessary background and motivation on random groups. This is joint work with Danny Calegari.

2PM, 129-104.

## November 4, Jason Behrstock (CUNY)

*Higher dimensional filling and divergence for mapping class groups*

We will discuss filling and divergence functions. We will describe their behaviors for mapping class groups of surfaces and show that these functions exhibit phase transitions at the rank, in analogy to the corresponding result for symmetric spaces. This work is joint with Cornelia Drutu.

## October 14, Michael Brandenbursky, (CRM, Univ. Montreal)

*Concordance group and stable commutator length in braid groups*

In this talk I will define quasi-homomorphisms from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, I will provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. I will also provide applications to the geometry of the infinite braid group. In particular, I will show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich. If time permits I will describe an interesting connection between the concordance group of knots and number theory. This work is partially joint with Jarek Kedra.

129-104

## October 7, Moon Duchin (Tufts)

*Geodesics in nilpotent groups*

Perhaps the simplest non-abelian infinite group to understand is the Heisenberg group H(Z). Given a generating set as our "alphabet," a geodesic in the group is an efficient "spelling" of a group element. It is quite challenging to understand these precisely for an arbitrary choice of generators, but the large-scale geometric structure of the group makes it possible.

129-104

## October 2, Sang-jin Lee (Konkuk Univ)

*Embedding of RAAG into braid groups*27-220

## September 30, Alessandro Sisto (ETH)

*Acylindrically hyperbolic groups*

Acylindrically hyperbolic groups form an extensive class of groups that contain, for example, non-elementary (relatively) hyperbolic groups, mapping class groups of hyperbolic surfaces and non-Abelian RAAGs. Their defining feature is that they admit a "non-trivial enough" action on some hyperbolic space. Despite the generality of the notion, many results can be proven about them, for example that they are SQ-universal, meaning that if G is acylindrically hyperbolic then any countable group embeds in some quotient of G (in particular, G has uncountably many pairwise non-isomorphic quotients). We will discuss geometric properties of acylindrically hyperbolic groups, focusing on ingredients that one can use to prove SQ-universality.

129-301

## August 7–12, Geometry on Groups and Spaces (ICM satellite)

### Plenary Talks

Click the titles for lecture videos.

Yves Benoist (Université Paris-Sud),

*Discrete subgroups of SL(3,R) containing nilpotent subgroups*Mladen Bestvina (University of Utah),

*Bounded cohomology with coefficients in uniformly convex Banach spaces*Martin Bridson (Oxford),

*Fibre Products, non-positive curvature, and profinite genera*Michelle Bucher-Karlsson (Université de Genève),

*Volume and characteristic numbers of representations of hyperbolic manifolds*(video missing)Marc Burger (ETH),

*On maximal representations of lattices in SU(1, n) into hermitian Lie groups*Richard Canary (U Michigan),

*Mapping Class Group Invariant Metrics on Higher Teichmuller Spaces*Ruth Charney (Brandeis),

*Morse Boundaries*Daryl Cooper (University of California, Santa Barbara),

*Convex Real Projective Structures*Patrick Foulon (Université de Strasbourg),

*Entropy and deformation of real projective structures on surfaces*Koji Fujiwara (Kyoto University),

*Quasi-homomorphisms into non-commutative groups*Ursula Hamenstädt (Universität Bonn),

*Convex cocompact subgroups of Out(Fn),*Alessandra Iozzi (ETH),

*Finite orbits and boundaries of CAT(0) cube complexes*Olga Kharlampovich (Hunter College, CUNY),

*Algorithmic problems for relatively hyperbolic groups and their limit groups*Thomas Koberda (Yale),

*Right-angled Artin groups and mapping class groups*Masato Mimura (Tohoku University),

*Group approximation in Cayley topology and coarse geometry*Shahar Mozes (Hebrew),

*Lattices in products of trees and a Theorem of H.C. Wang*Denis Osin (Vanderbilt),

*New examples of C*-simple groups*Joan Porti (Universitat Autònoma de Barcelona),

*Coarse geometry of isometry groups of symmetric spaces*Alan Reid (University of Texas at Austin),

*The geometry of canonical curves*Mark Sapir (Vanderbilt),

*Tarski numbers of groups*Karen Vogtmann (Cornell),

*Building cocycles for Out(Fn)*Dani Wise (McGill),

*Cubulating Free-by-cyclic groups'**'*

### Long Sessions

Thierry Barbot (Avignon),

*Construction of flat spacetimes in expansion with particles*Ara Basmajian (City Univ. of New York),

*Hyperbolic surface identities*Suhyoung Choi (KAIST),

*The convex real projective orbifolds with radial or totally geodesic ends: The closedness and openness of deformations*Francois Dahmani (Institut Fourier, Grenoble),

*Conjugacy problem for some automorphisms of free groups*Pallavi Dani (Louisiana State),

*Quasi-isometry and commensurability for right-angled Coxeter groups*Sergio Fenley (Florida State Univ),

*Closed orbits of pseudo-Anosov flows: cardinality and length growth*Daniel Groves (Univ. of Illinois at Chicago),

*The Malnormal Special Quotient Theorem*Martin Kassabov (Cornell),

*Hopf algebras and invariants of the Johnson cockerel*Fanny Kassel (CNRS & Universite Lille 1),

*Anosov representations and proper actions*Thilo Kuessner (KIAS),

*Proportionality principle for noncompact manifolds*Eiko Kin (Osaka Univ.),

*Dynamics of the monodromies of the fibrations on the magic 3-manifold*Seonhee Lim (Seoul National Univ.),

*Martin boundary and Brownian motion on hyperbolic manifolds*Sara Maloni (Brown),

*Polyhedra inscribed in quadrics, anti-de Sitter and half-pipe geometry*Jason Manning (Cornell),

*Dehn filling and ends of groups*Eduardo Martinez-Pedroza (Memorial),

*On Subgroups of Non-positively Curved Groups*Hiroshige Shiga (Tokyo Institute of Technology),

*Teichmüller curves and holomorphic maps on Riemann surfaces*Stephan Tillmann (The Univ. of Sydney),

*Thurston norm via Fox calculus*Andrey Vesnin (Sobolev Institute of Mathematics),

*On Jorgensen numbers of hyperbolic 3-orbifold groups*Henry Wilton (Cambridge),

*3-manifolds in random groups*Genkai Zhang (Chalmers University of Technology and Gothenburg University),

*Kähler metric over Hitchin component*

### Short Sessions

Ilesanmi Adeboye (Wesleyan),

*The Area of Projective Surfaces*Juan Alonso (Universidad de la Republica - Uruguay),

*Measure free factors of free groups*Yago Antolin (Vanderbilt),

*Finite generating sets of relatively hyperbolic groups*Lucien Clavier (Cornell),

*Geometric limits of cyclic subgroups of PSL2(R)*Matthew Durham (Univ. Illinois at Chicago),

*Elliptic Actions on Teichmüller Space*Steven Frankel (Yale),

*Quasigeodesic flows and dynamics at infinity*Simion Filip (Univ. of Chicago),

*Hodge theory and rigidity in Teichmuller dynamics*David Hume (Universite Catholique de Louvain),

*Group approximation in Cayley topology and coarse geometry*Woojin Jeon (KIAS),

*Some applications of Cannon-Thurston map*Jingyin Huang (New York Univ.),

*Quasi-isometry classification of right-angled Artin group with finite outer automorphism group*Sungwoon Kim (KIAS),

*Simplicial volume and its applications*Gye-Seon Lee (Univ. of Heidelberg),

*Andreev’s theorem on projective Coxeter polyhedra*HyoWon Park (UNiv. of Utah),

*A metric on an outer space for 2-dimensional right-angled Artin groups*Catherine Pfaff (Universite d'Aix-Marseille),

*A Dense Geodesic Ray in Certain Subcomplexes of Quotiented Outer Space*Jaipong Pradthana (Chiang Mai University),

*Totally geodesic surfaces in closed hyperbolic 3-manifolds*Alessandro Sisto (ETH),

*Bounded cohomology of acylindrically hyperbolic groups*Alden Walker (Univ. Chicago),

*Homologically essential surface subgroups of random groups*Maxime Wolff (Paris 6),

*The modular action on PSL(2,R)-characters in genus two*Tengren Zhang (Univ. of Michigan),

*Degeneration of convex real projective structures on surfaces*

# 2013

Geometric group seminar at KAIST.

## August 12- 16, KAIST Geometric Topology Fair

### Lecture series

Mladen Bestvina (University of Utah, USA),

*The geometry of mapping class groups and Out(Fn)*Dave Witte Morris (University of Lethbridge, Canada),

*Arithmetic subgroups of SL(n,R)*Joan Porti (Universitat Autònoma de Barcelona, Spain),

*Dynamics at infinity of symmetric spaces*

### Research talks

Delaram Kahrobaei (Graduate Center, CUNY),

*Polycyclic groups: A Secure Platform for Ko-Lee Protocol*Youngju Kim (KIAS),

*Quasiconformal deformations of Schottky groups in complex hyperbolic space*Thilo Kuessner (KIAS),

*Proportionality principle for simplicial volume*Sangyop Lee (Chung-Ang University),

*Twisted torus knots*Jung Hoon Lee (Chonbuk National University),

*Topologically minimal surfaces*Seonhee Lim (Seoul National University),

*Subword complexity and Sturmian colorings of trees*Ken'ichi Ohshika (Osaka U, Japan),

*Geometric limits and deformation spaces of Kleinian groups*Catherine Pfaff (Universite Aix-Marseille, France),

*Stratifying the set of fully Irreducible elements of Out(Fr)*

## July 9 - 11, Thomas Koberda (Yale)

Two lectures on

*curve graphs for right-angled Artin groups*Lecture I. Curve graphs for right-angled Artin groups I: right-angled Artin group actions on the extension graph

I will discuss basics of extension graphs for right-angled Artin groups and the actions of right-angled Artin groups on their extension graphs. The main result in this lecture will be a version of the Nielsen--Thurston classification for right-angled Artin groups. Joint with S. Kim.

Lecture II. Curve graphs for right-angled Artin groups II: the large and small scale geometry of the extension graph

I will discuss various aspects of the geometry of the extension graph. I will discuss vertex link projections, the bounded geodesic image theorem, and a distance formula for right-angled Artin groups. Joint with S. Kim.

July 9 and 11, 4-5 pm. Building E6-1, Room 4415.

## May 20 - June 10, Igor Mineyev (UIUC)

Ten lectures on

*Groups, Cell Complexes and l2-Homology*

One of the longest outstanding conjectures in group theory was Hanna Neumann Conjecture (1957):

if H and K are subgroups of a free group, then rank(H∩K)-1 ≤ (rank(H)-1) (rank(K)-1).

A long list of group theorists attempted to prove this conjecture, and also have contributed to related results. Mineyev first resolved this question completely in a paper at Annals of Mathematics (2012). In this series of ten lectures, he will give the main construction of the proof and its applications.

Topics

Cell Complexes and Group Actions.

The L2 Homology and L2 Betti Numbers.

The Hanna Neumann Conjecture.

Orderability of Groups.

Submultiplicativity and the Deep-Fall Property.

The Atiyah Problem.

May 20 - June 10. MWF 11 - 12:30 pm. Building E6-1, Room 2411.

One-credit intensive course (MAS 583).

## May 30, Irene Peng (POSTECH)

4:30 - 5:30 pm. Room 1501.

*Amenability and all that*

Amenability is one of those properties of group that has many different characterizations. I will discuss what it means in terms of invariant means, random walks and C* algebras. If time permits, I will also describe some related notions such as property rapid decay in the C* algebra setting.

## June 4, Hyungryul Baik (Cornell)

4:30 pm - 5:30 pm. Room 4415.

*Circular-Orderability of Three-Manifold Groups and Laminations of the Circle*

We will discuss the connection between the circular-orderability of the fundamental group of a 3-manifold M and the existence of certain codimension-1 foliations on M via Thurston's universal circle theory. This theory provides a motivation to study group actions on the circle with dense invariant laminations. As an one lower dimensional example, we will give a complete characterization of Fuchsian groups in terms of its (topological) invariant laminations.

## April 10 - 15, Jason Fox Manning (U of Buffalo)

Four Lectures on

*Hyperbolic Dehn Fillings of Groups and Spaces*Lecture 1: The Gromov-Thurston 2\pi Theorem.

In the first lecture, I'll describe an explicit construction of negatively curved metrics on closed 3-manifolds obtained by Dehn filling of cusped hyperbolic manifolds. I also plan to sketch an application by Cooper and Long to finding surface subgroups of 3-manifolds. I'll talk about how to extend the 2\pi Theorem to cusped hyperbolic manifolds of dimension larger than 3.

Lecture 2: Relatively hyperbolic groups.

I'll define and give examples of relatively hyperbolic groups, and talk about what it means to do Dehn filling on a group pair.

Lecture 3: The relatively hyperbolic Dehn filling theorem.

I'll state the main theorem and sketch a proof.

Lecture 4: Quasiconvex subgroups and Dehn filling.

I'll define relatively quasiconvex subgroups, and talk about how to do Dehn filling while preserving quasiconvexity.

April 10 (W), 11 (Th), 12 (Fr), 16 (T). 4 pm - 5:15 pm. Room 4415.

## May 3 - 9, Cameron McA. Gordon (U Texas at Austin)

Four Lectures on

*Dehn Surgery and Three-Manifold Groups*Seminar: Dehn Surgery

The lectures will be an introduction to Dehn surgery. This is a construction, going back to Dehn in 1910, for producing closed 3-manifolds from knots. A natural generalization is Dehn filling, in which some torus boundary component $T$ of a 3-manifold $M$ is capped off with a solid torus $V$. If $\alpha$ is the isotopy class of the loop on $T$ that bounds a disk in $V$, the resulting filled manifold is denoted by $M(\alpha)$. Generically, the topological and geometric properties of $M$ persist in $M(\alpha)$; in particular if $M$ is hyperbolic then $M(\alpha)$ is usually also hyperbolic. If this fails then the filling is said to be {\it exceptional}. We will outline a program to classify the triples $(M;\alpha,\beta)$ with $M(\alpha)$ and $M(\beta)$ exceptional, describing what is known in this direction and what remains to be done.

Colloquium: Left-Orderability of Three-Manifold Groups

We will discuss connections between three notions in 3-dimensional topology that are, roughly speaking, algebraic, topological, and analytic. These are: the left-orderability of the fundamental group of a 3-manifold M, the existence of certain codimension 1 foliations on M, and the Heegaard Floer homology of M.

Reference

Park City Lectures Dehn Surgery and 3-Manifolds, in Low Dimensional Topology, ed. T.S. Mrowka and P.S. Ozsvath, IAS/Park City Mathematics Series, Vol. 15, AMS 2009.

Seminar May 3 (F), 7 (T), 8 (W). 4:30 - 5:30 pm. Room 4415

Colloquium May 9 (Th). 4:30 - 5:30 pm. Building E6, 1501.

# 2012

At KAIST.

## August 13 - 17, the 10th KAIST Geometric Topology Fair

### Lecture Series

Michael Davis (Ohio State Univ),

*Graph products, RACGs and RAAGs*Koji Fujiwara (Tohoku Univ),

*Group actions on quasi-trees*Alan Reid (Univ of Texas at Austin),

*3-manifold groups, covering spaces and LERF*

### Research Talks

Jinseok Cho (KIAS),

*Introduction to Kashaev volume conjecture*Kanghyun Choi (KAIST),

*The definability criterion for cocompact convex projective polyhedral reflection groups*Stefan Friedl (Univ. of Cologne),

*Minimal genus surfaces in 4-manifolds with a free circle action*Thilo Kuessner (KIAS),

*Invariants preserved by mutation*Sang-hyun Kim (KAIST),

*Hyperbolic aspects of right-angled Artin groups*Taehee Kim (Konkuk Univ.),

*Knot concordance and invariants from derived covers*Sang-Jin Lee (Konkuk Univ.),

*Braid group of type (de,e,r)*Gye-Seon Lee (Seoul National Univ.),

*Real projective deformations of hyperbolic reflection orbifolds*Seonhee Lim (Seoul National Univ.),

*Ford circles and Farey maps for function field*Ken’ichi Ohshika (Osaka University),

*Actions of isomorphism groups of Heegaard splittings on projective lamination spaces*

## August 21 - 30, Genevieve S. Walsh (Tufts)

Four Lectures on

*Introduction to Hyperbolic Orbifolds and Knot Commensurability*.Aug 21, 23, 28, 30. TTh 4 - 5 pm. Room 3433.

Lecture 1: 2-dimensional orbifolds

In this lecture we will define and describe orbifolds and set notation. In particular, we will discuss orbifold Euler characteristic, orbifold covers, good orbifolds, bad orbifolds, and the orbifold fundamental group. Explicit examples of spherical, Euclidean and hyperbolic 2-orbifolds will be given. We will also prove that there is a smallest closed hyperbolic 2-orbifold.

Lecture 2: 3-dimensional orbifolds

Here we will explore 3-dimensional orbifolds, restricting mainly to good orbifolds. Although we will give explicit examples of many different types of 3-dimensional orbifolds, the focus will be on hyperbolic 3-orbifolds. To this end, we will discuss hyperbolic isometries and the geometry of hyperbolic orbifolds and hyperbolic orbifolds. We will discuss how useful orbifolds are to the study of 3-manifolds, and give a statement of geometrization.

Lecture 3: Commensurability

Commensurability is an equivalence relation on manifolds and orbifolds which is a refinement of geometrization. Here we will describe the current study of commensurability of hyperbolic manifolds, focusing on commensurability of knot complements. We will describe hyperbolic knot complements and their symmetry groups, and discuss the commensurator group and the orbifold commensurator quotient of a hyperbolic non-arithmetic knot complement.

Lecture 4: Some results on commensurability of knot complements

A conjecture of Reid and Walsh asserts that there are at most 3 hyperbolic knot complements in any commensurability class. Here we discuss this conjecture, and give results under certain circumstances. The problem naturally divides itself into two cases, the case of hidden symmetries and the case of no hidden symmetries, and we discuss both. The new results presented here are joint with M. Boileau, S. Boyer, and R. Cebanu.

References

Not Knot Part 1 (video)

Not Knot Part 2 (video)

Thurston's course notes, Chapter 13

Michael Kapovich, Hyperbolic manifolds and discrete groups, Chapter 6

PDF "Orbifolds and commensurability"

## September 5–7, Ian Agol (UC Berkeley)

Three Lectures on

*Virtual Haken Conjecture*.September 5, 6, 7 (WThF), 4 - 5 pm. Room 3433.

Prequel Lecture September 5, Wednesday, 4 – 5 pm @ Room 3433

An Invitation to Cube Complexes by Sang-hyun Kim (Slides)

We survey basic facts on cube complexes and discuss how those facts are related to the study of subgroups of right-angled Artin groups.

Lecture I

We will discuss the proof of the virtual Haken conjecture and related questions. The first lecture will be an overview and an explanation of how to reduce the problem to a conjecture of Wise in geometric group theory.

Lecture II

The second lecture will be on the RFRS condition and virtual fibering for hyperbolic 3-manifolds.

Lecture III

The third lecture will be on the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special.

The following video lectures @ KAIST by Alan Reid are not prerequisites, but provide valuable information on the background and a big picture surrounding this problem:

*3-manifold groups, covering spaces and LERF*

http://vod.mathnet.or.kr/sub2_2.php?no=2437 http://vod.mathnet.or.kr/sub2_2.php?no=2445 http://vod.mathnet.or.kr/sub2_2.php?no=2450

Selected References

F. Haglund and D. Wise, Special cube complexes, Geom. Funct. Anal. (2007) 1–69.

I. Agol, Criteria for virtual fibering, J. Topol. 1 (2008), no. 2, 269–284.

D. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy., Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.

CBMS lecture videos, notes and related preprints: http://comet.lehman.cuny.edu/behrstock/cbms/program.html

## September 25–October 4, Thomas Koberda (Yale)

Four Lectures on

*Mapping Class Groups and Right-Angled Artin Groups*

*An Introduction to Right-Angled Artin Groups and Mapping Class Group*In this lecture, we will begin with some basic facts about right-angled Artin groups and mapping class groups. The goal is to provide a foundation for various new results concerning the structure and geometry of right-angled Artin groups, mapping class groups, and their subgroups.*An Introduction to Right-Angled Artin Groups and Mapping Class Groups*In this lecture, we will discuss the primary result of [3], which roughly says that if we take any collection of mapping classes, say {f1,...,fk} and replace them by sufficiently high powers {f1^N,...,fk^N}, they generate a right-angled Artin subgroup of the mapping class group of the expected type. Unless otherwise noted, all examples and statements can be found with proof (or appropriate reference) in [3].*Right-Angled Artin Subgroups of Right-Angled Artin Groups*In this lecture, we will discuss the primary results of [2]. In that article, the authors develop a general theory for determining when there exists an embedding A(X) -> A(Y) for two graphs X and Y.*A Dictionary Between Mapping Class Groups and Right-Angled Artin Groups Via Curve Complexes*In this lecture, we will primarily be discussing the results of [1], together with appropriate background. The general principle we would like to explore is that right-angled Artin groups behave a lot like mapping class groups from the point of view of their actions on their extension graphs and curve complexes respectively.

References

Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.

Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.

Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.

September 25, 27, October 2, 4. T 4 - 5 pm, Th 2:30 - 3:30. Room 3433

# 2011

At KAIST.

## August 8–12, The 9th KAIST Geometric Topology Fair

### Lecture Series

Danny Calegari (Caltech),

*Random rigidity in free groups*Jason F. Manning (SUNY Buffalo),

*Recognizing low-dimensional manifold groups*Mark V. Sapir (Vanderbilt),

*Embedding finitely generated groups into finitely presented groups*Efim Zelmanov (UC San Diego),

*Asymptotic group theory - pro-p group; property T and expanders*

### Research Talks

Sungwoon Kim (KIAS),

*Simplicial volume and bounded group cohomology*Sang-hyun Sam Kim (KAIST),

*Embeddability between right-angled Artin groups*Kihyoung Ko (KAIST),

*Graph braid groups: its 10 year history*Donghi Lee (Pusan National University),

*Combinatorial group theory applied to 2-bridge link groups*Seonhee Lim (Seoul National University),

*Commensurizer group and its growth*

## September 1 - 16, Jon McCammond (UC Santa Barbara)

Ten Lectures on

*Coxeter Groups and Reflection Symmetry*September 1 - 16, 2011 (except for Saturday, Sunday and 09/12, 09/13) MWF 4 - 5:30 pm, TTh 1 - 2:30 pm

One-credit intensive course.

Symmetry and Abstraction -

*Why do mathematicians see only 17 types of wallpaper?*, a public Lecture

Coxeter groups are a central object of study in many parts of mathematics. They include the groups of symmetries of the regular polytopes, the finite reflection groups and the Weyl groups at the core of the study of Lie groups and Lie algebras. They have many remarkable properties including the fact that they have faithful linear representations and a proper cocompact action by isometries on piecewise Euclidean space of nonpositive curvature. In this course I will focus on laying the foundations for the geometry, topology and combinatorics of Coxeter groups.

Prerequisites

The prerequisites are only Linear Algebra and Abstract Algebra (i.e. groups). Some familiarity with groups given by generators and relations, and fundamental groups and covering spaces would be nice but probably not absolutely necessary.

Lecture Plan

1 & 2 Regular polytopes and spherical Coxeter groups

3 & 4 Lie groups and Euclidean Coxeter groups

5 & 6 Coxeter groups in general

7 & 8 Linear representations and basic facts

9 & 10 Non-positively curved spaces and geometric actionsGrading

Letter grades are given based on (1) attendance / participation (2) a short paper due 09/01/2011.