Interactions between low dimensional topology and mapping class groups

July 1-5, 2013, Max Planck Institute for Mathematics, Bonn

Organizing Committee

R. Inanc Baykur (Max Planck Institute), John Etnyre (Georgia Institute of Technology) and Ursula Hamenstädt (University of Bonn)

Invited speakers and participants

  • Tara Brendle (University of Glasgow)
  • Martin Bridson (University of Oxford)
  • Hisaaki Endo (Tokyo Institute of Technology)
  • Stefan Friedl (University of Cologne)
  • Daniel Groves (University of Illinois at Chicago)
  • Kenta Hayano (Osaka University)
  • Jonathan Hillman (University of Sydney)
  • Keiko Kawamuro (University of Iowa)
  • Will Kazez (University of Georgia)
  • Sang-hyun Kim (KAIST)
  • Thomas Koberda (Yale University)
  • Mustafa Korkmaz (Middle Eastern Technical University)
  • Paolo Lisca (Universitŕ di Pisa)
  • Gregor Masbaum (Institut de Mathematiques de Jussieu)
  • Patrick Massot (Université Paris Sud)
  • Gordana Matic (University of Georgia)
  • Naoyuki Monden (Kyoto University)
  • Burak Ozbagci (Koc University)
  • Andy Putman (Rice University)
  • Alan Reid (University of Texas at Austin)
  • Lee Rudolph (Clark University)
  • Andras Stipsicz (Alfréd Rényi Institute)
  • Jeremy Van Horn-Morris (University of Arkansas)
  • Karen Vogtmann (Cornell University)
  • Andy Wand (Harvard University)

Goals

There has been a long history of rich and subtle connections between low dimensional topology, mapping class groups and geometric group theory. The goal of this workshop is to highlight these connections and foster new and unexpected collaborations between researchers in these areas. This will be done through a combination of research talks, longer lectures and some ample time to engage with other workshop participants.

Registration and support

[Funding applications are now closed.]

Accommodation for invited participants will be provided by the Max Planck Institute. We ask all the participants to register at this link. For funding, you should apply and arrange your recommendation letters to be sent to us by May 15, 2013. We particularly encourage graduate students and recent PhDs to apply. For further inquiries regarding accommodation contact [housing "AT" mpim-bonn.mpg.de].

The conference is funded by the Max Planck Institute for Mathematics and the Haussdorff Center for Mathematics in Bonn.

Schedule

Monday, July 1

08:15-09:00 Registration
09:00-10:30 Stipsicz
10:30-11:00 TEA
11:00-12:30 Vogtmann

14:00-14:50 Hillman
15:10-16:00 Koberda
16:00-16:30 TEA
16:30-17:20 Friedl
Tuesday, July 2


09:00-10:30 Reid
10:30-11:00 TEA
11:00-12:30 Massot

14:00-14:50 Masbaum
15:10-16:00 Wand
16:00-16:30 TEA
16:30-17:20 Wand
17:30-19:30 Reception
Wednesday, July 3


09:00-10:30 Etnyre
10:30-11:00 TEA
11:00-12:30 Bridson

14:00-14:50 Van Horn-Morris
15:10-16:00 Kazez
16:00-16:30 TEA
16:30-17:20 Lisca
Thursday, July 4


09:00-10:30 Groves
10:30-11:00 TEA
11:00-12:30 Rudolph

14:00-14:50 Kawamuro
15:10-16:00 Putman
16:00-16:30 TEA
16:30-17:20 Brendle
Friday, July 5


09:00-10:30 Korkmaz
10:30-11:00 TEA
11:00-12:30 Endo
14:00-14:50 Ozbagci
15:00-15:30 TEA

All talks will be held at the MPIM Lecture Hall (3rd floor), tea and the Tuesday Reception at the MPIM Tea Room (4th floor). Directions to the Institute can be found here.

Titles and abstracts of talks

  • András Stipsicz
    Symplectic 4-manifolds and Lefschetz fibrations
    In the survery we review the basic features of symplectic 4-manifolds which are relevant in the study of mapping class groups. We list some problems which can be verified using methods coming from symplectic topology and show how theorems of, say, Seiberg-Witten theory can be applied in their resolution.

  • Karen Vogtmann
    Automorphism groups of right-angled Artin groups
    Right-angled Artin groups (RAAGs) interpolate between free groups and free abelian groups, and one may view their outer automorphism groups as interpolating between Out(F_n) and GL(n,Z). After summarizing what is known about these automorphism groups I will explain how to build contractible spaces on which they act properly. For free abelian groups this is the usual symmetric space with its GL(n,Z) action, and for free groups this is Outer space, with its action of Out(F_n).

  • Jonathan Hillman
    4-Manifolds and Surface Bundles
    The total spaces of bundles with base and fibre aspherical closed surfaces are determined up to TOP s-cobordism by their fundamental groups and Euler characteristics. A given bundle space may admit more than one bundle projection, but there are only finitely many which are essentially distinct. We shall sketch (briefly) the arguments, consider the extension to 4-manifolds which fibre over aspherical 2-orbifolds, and raise some questions.

  • Thomas Koberda
    Masur-Minsky style curve complex machinery for right-angled Artin groups
    I will explain joint work with Sang-hyun Kim, in which we develop curve complex machinery for right-angled Artin groups. The central results are the acylindricity of the right-angled Artin group action on the analogue of the curve graph, and a bounded geodesic image theorem. From these results, we can deduce verbatim analogues of many theorems about mapping class groups in the context of right-angled Artin groups.

  • Stefan Friedl
    The virtual fibering theorem for 3-manifolds
    We will give an alternative account of Agol's proof that 3-manifolds with RFRS fundamental group are virtually fibered. Together with the recent work of Agol, Przytycki-Wise and Wise this implies that 'most' 3-manifolds are virtually fibered. This is joint work with Takahiro Kitayama.

  • Alan Reid
    Groups in which all finite groups are involved
    Let G and H be groups, say that H is involved in G if there is a finite index subgroup K in G such that K surjects onto H. In this talk we discuss a construction to gaurantee that all finite groups are involved in a certain class of groups. This includes the Mapping Class groups, and uses the projective unitary representations arising from TQFT (which will be discussed in Masbaum's talk).

  • Patrick Massot
    Contact structures and open books on 3-manifolds
    In this lecture I will describe the Giroux correspondence relating contact structures on 3-manifolds and surface mapping classes (or, more precisely, open book decompositions for 3-manifolds). I will not assume any background in contact topology.

  • Gregor Masbaum
    Integral TQFT and modular representations of mapping class groups
    This talk will be about modular representations in finite characteristic of mapping class groups of surfaces coming from the theory of Integral SO(3) Topological Quantum Field Theory. These representations were used in joint work with Reid which will be presented in Reid's talk at this conference. Here, we will give some more information about these representations, with particular focus on the case of so-called equal characteristic.

  • Andy Wand
    Tightness and open book decompositions (part I)
    A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a stabilization-invariant property of open books which corresponds to tightness of the corresponding contact structure. We will mention applications to the classification of contact 3-folds, and also to the question of whether tightness is preserved under Legendrian surgery.

    Tightness and open book decompositions (part II)
    This second talk will be a somewhat more technical exposition of the arguments and methods introduced in Part I.

  • John Etnyre
    Submonoids of mapping class groups and contact topology
    Giroux's correspondence relates open book decompositions of a manifold M to contact structures on M. This correspondence has been fundamental to our understanding of contact geometry. An intriguing question raised by this correspondence is how geometric properties of a contact structure are reflected in the monodromy map describing the open book decomposition. In this talk I will show that there are several interesting monoids in the mapping class group that are related to various properties of a contact structure (like being Stein fillable, weakly fillable, and so on) and discuss what is known about them. This is joint work with Jeremy Van Horn-Morris and Ken Baker.

  • Martin Bridson
    Non-positive curvature, cubes and subgroups of mapping class groups
    In this talk I shall discuss the subgroup structure of mapping class groups, with a particular emphasis on constructions that involve non-positively curved cubed complexes.

  • Jeremy Van Horn-Morris
    Positive factorizations of mapping classes and the topology of Stein fillings
    The Giroux correspondence allows us to encode a contact 3-manifold as an open book decomposition, and hence as the mapping class of some surface diffeomorphism. In this setup, certain nice symplectic 4-manifolds which fill the 3-manifold arise as factorizations of the mapping class element into right handed Dehn.twists. I'll discuss some of the difficulties of the technology, as well as what we know and don't know about the general behaviors exhibited by open books and their positive factorizations. This is joint with I. Baykur.

  • Will Kazez
    An overview of fractional Dehn twisting
    Given a homeomorphism of a surface with boundary, fractional Dehn twisting measures the difference, at the boundary, of the homeomorphism and its pseudo-Anosov representative. I will give a survey of this concept starting with the work of Gabai in the study of essential laminations and continuing with the role it plays in contact topology.

  • Paolo Lisca
    Stein fillable contact 3--manifolds and positive open books of genus one
    A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, Wand as well as Baker, Etnyre and Van Horn-Morris constructed counterexamples to the converse statement with S of genus two. In this talk I will present a proof of the converse statement under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L-space. If time permits I will describe a (still conjectural) classification up to diffeomorphisms of the Stein fillings of (Y(S,h), C(S,h)), where S is a one-holed torus, h is positive and Y(S,h) is a Heegaard Floer L-space.

  • Daniel Groves
    The Malnormal Special Quotient Theorem
    The Malnormal Special Quotient Theorem of Wise is the technical result in his work on hyperbolic groups with a quasiconvex hierarchy, and also a key tool in Agol's proof of the Virtual Haken Conjecture. In this talk, I will explain where the MSQT fits into this story, and outline a new proof due to Agol, Manning and myself. This is joint work with Ian Agol and Jason Manning.

  • Lee Rudolph
    Quasipositive braids and complex geometry
    First I will review some known applications of the notion of "quasipositivity" in braid theory, knot theory, and elsewhere in low-dimensional topology, e.g., the theories of contact structures, open books, Lefschetz pencils, and Stein surfaces. Then I will describe some places where I suspect (hopefully) that unknown applications---either of or to quasipositivity---may be waiting to become known, e.g., commutator lengths in braid groups, invariants of transverse C-links, and the topology of spaces of pieces of complex curves.

  • Keiko Kawamuro
    The self linking number of transverse links and the Johnson-Morita homomorphism
    The self linking number is a classical invariant of transverse knots in contact 3 manifolds. By Bennequin, Mitsumatsu-Mori, and Pavelescu, a transverse link can be identified with a closed braid relative to an open book decomposition supporting the ambient contact structure. Bennequin found a self-linking formula of transverse links in the standard contact 3-sphere. His formula is written in terms of the braid index and the exponent sum of a braid word. In this talk I will present a self-linking formula of transverse links in general contact manifolds. The formula contains a new term involving the Johnson-Morita homomorphism of the monodromy of an open book supporting the contact structure. This is joint work with Tetsuya Ito.

  • Andy Putman
    Generating the Johnson filtration
    The kth term of the Johnson filtration of the mapping class group of a surface S, denoted I(S,k), is the kernel of the action of the mapping class group on the kth nilpotent truncation of the surface group (for k=1, this is the Torelli group). Work of Birman-Powell and Johnson gives generators for I(S,1) and I(S,2), respectively; however, generators are not known for I(S,k) for k>2. I will discuss a proof that for all k, there exists some n_k such that I(S,k) is generated by elements supported on subsurfaces of genus at most n_k. The proof uses recent work concerning the representation theory of the symmetric group, in particular the notions of *central stability* and *FI-modules* which were introduced by myself and Church-Ellenberg-Farb, respectively. This is joint work with Tom Church.

  • Tara Brendle
    Combinatorial properties of the kernel of the integral Burau representation
    We will describe a simple generating set for three related groups: the kernel of the integral Burau representation, the hyperelliptic Torelli group, and the fundamental group of the branch locus of the period mapping. This result confirms a conjecture of Hain (this is joint work with Dan Margalit and Andy Putman). We will also explain an algorithmic approach for factoring a wide class of elements of these groups in terms of this generating set, leading to interesting and elementary new relations (joint with Dan Margalit).

  • Mustafa Korkmaz
    Commutator lengths in mapping class groups
    I will first give definitions and some properties of commutator lengths, stable commutator lengths, bounded cohomology and the relations between them. I will then give a survey of known results about the commutator lengths of elements (mostly Dehn twists) in the mapping class group of an orientable surface.

  • Hisaaki Endo
    Lefschetz fibrations and the signature cocycle
    In this talk I first recall definitions and basic properties of Lefschetz fibrations and their monodromy. I next review the signature cocycle discovered by Meyer and its relation to the ternary Maslov index. Then I discuss signature computations for Lefschetz fibrations and related topics, for example, chart description of Lefschetz fibrations.

  • Burak Ozbagci
    Symplectic fillings of lens spaces as Lefschetz fibrations
    We construct a positive allowable Lefschetz fibration over the disk on any minimal symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity. (This is a joint work with Mohan Bhupal.)