Interactions between low dimensional topology and mapping class groupsJuly 1-5, 2013, Max Planck Institute for Mathematics, Bonn
Organizing CommitteeR. Inanc Baykur (Max Planck Institute), John Etnyre (Georgia Institute of Technology) and Ursula Hamenstädt (University of Bonn)
Invited speakers and participants
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.
Monday, July 1
Tuesday, July 2
Wednesday, July 3
14:00-14:50 Van Horn-Morris
Thursday, July 4
Friday, July 5 |
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
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.
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).
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.)