Geometric group theory seminar@SNU (Past)

For current seminar, please visit http://seminar.cayley.kr

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.

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

2013

Geometric group seminar at KAIST.

August 12- 16, KAIST Geometric Topology Fair

• The 11th KAIST Geometric Topology Fair
• Co-organizers: Sang-hyun Kim, Suhyoung Choi and Kihyoung Ko

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, KAIST Geometric Topology Fair

• 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 be aimed at graduate students and early career researchers.
• Lecture Series by: Michael Davis (Ohio State Univ), Koji Fujiwara (Tohoku Univ.), Alan Reid (Univ. Texas at Austin).

Research Talks by: Jinseok Cho (KIAS), Kanghyun Choi (KAIST), Stefan Friedl (Univ. of Cologne), Thilo Kuessner (KIAS), Sang-hyun Kim (KAIST), Taehee Kim (Konkuk Univ.), Sang-Jin Lee (Konkuk Univ.), Gye-Seon Lee (Seoul National Univ.), Seonhee Lim (Seoul National Univ.), Ken’ichi Ohshika (Osaka University).

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
1. Not Knot Part 1 (video)
2. Not Knot Part 2 (video)
3. Thurston's course notes, Chapter 13
4. Michael Kapovich, Hyperbolic manifolds and discrete groups, Chapter 6
5. 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
• Selected References
1. F. Haglund and D. Wise, Special cube complexes, Geom. Funct. Anal. (2007) 1–69.
2. I. Agol, Criteria for virtual fibering, J. Topol. 1 (2008), no. 2, 269–284.
3. D. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy., Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.
4. 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
1. 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.
2. 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].
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.
4. 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
1. Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.
2. Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.
3. 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

Random rigidity in free groups, Danny Calegari (Caltech)
Recognizing low-dimensional manifold groups, Jason F. Manning (State University of New York at Buffalo)
Embedding finitely generated groups into finitely presented groups, Mark V. Sapir (Vanderbilt)
Asymptotic group theory - pro-p groups; property T and expansion, Efim Zelmanov (Univeristy of California at San Diego)
• Research Talks
Simplicial volume and bounded group cohomology, Sungwoon Kim (KIAS)
Embeddability between right-angled Artin groups, Sang-hyun Kim (KAIST)
Graph braid groups: its 10 year history, Kihyoung Ko (KAIST)
Combinatorial group theory applied to 2-bridge link groups, Donghi Lee (Pusan National University)
Commensurizer group and its growth, Seonhee Lim (Seoul National University)

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.