Talks

Alexandr Bufetov
Limit theorems for parabolic flows (figures1, figures2, figures3, figures4)

The  main results of the talk are limit theorems for two classes of measure-preserving flows of geometric origin.
The first class is given by translation flows on flat surfaces. The second class, for which the result re joint with Giovanni Forni, consists of horocycle flows on compact surfaces of constant negative curvature.
One of the main objects introduced in the talk is a special family of horocycle-invariant finitely-additive Hoelder measures on rectifiable arcs. An asymptotic formula for ergodic integrals of our flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula.

Pierre Derbez
Chern Simons Theory and the volume  of three-manifolds (slides)

We give some applications of the Chern-Simons gauge theory to the study of the set vol(N,G) of volumes of all representations ρ : π1 N G, where N is a closed oriented 3-manifold and G is either ${\rm Iso}_e\widetilde{\rm SL_2(\mathbb R)}$ or Iso+H3. We will focus on the following questions:
(1) How to find non-zero values in vol(N, G)? or weakly how to find non-zero elements in vol(Ñ, G) for some finite cover Ñ of N?
(2) What kind of topological information is enclosed in the elements of vol(Ñ, G)?
(3) Do these volumes satisfy the covering property in the sense of Thurston?
This is joint work with Shicheng Wang.

François Guéritaud
Lorentzian manifolds with constant curvature (slides)

PSL2(R), endowed with its Killing form, can be seen as the model space for negatively curved Lorentzian geometry. Its group structure permits a handy description of its proper quotients: they are given by pairs of reprensentations of a Fuchsian group, acting diagonally by left and right multiplication, with the property that one representation moves points of H2 "less" than the other (Kassel). I will present an infinitesimal analogue, which lets proper affine quotients of Minkowski 3-space (or "Margulis spacetimes") appear as geometric limits of PSL2 quotients. In particular, this implies Margulis spacetimes are topologically handlebodies. Joint work with F. Kassel and J. Danciger.

Kazuo Habiro
Unified quantum invariants of integral homology 3-spheres associated to simple Lie algebras (slides)

The Witten-Reshetikhin-Turaev quantum 3-manifold invariants τMg(ζ) are defined for finite-dimensional simple Lie algebras g and roots of unity ζ.
For an integral homology sphere M, we define a "unified invariant" Jmg, taking values in a completion of the polynomial ring Z[q], which is called the cyclotomic completion of Z[q] or the ring of analytic functions on the set of roots of unity.
The quantum invariants τMg(ζ) at various roots of unity recover from this unified invariant by setting q=ζ . This is a joint work with Thang Le, and generalizes my previous result for the sl2 case.

Kentaro Ito
On the topology of the space of Kleinian once-punctured torus groups (slides)

It is known that the topology of the space of Kleinian once-punctured torus groups is fairly complicated; it self-bumps and, moreover, it is not locally connected (due to Bromberg).
In this talk we first characterize sequences of punctured torus groups which induce self-bumping of the space.
We next explain Bromberg's theory from the viewpoints of the trace coordinate and the complex Fenchel-Nielsen coordinate.
Especially, the shape of linear slices which are close to the Maskit slice will be discussed.

Nariya Kawazumi

The completed Goldman-Turaev Lie bialgebra and mapping class groups (slides)

We introduce a completion of the Goldman-Turaev Lie bialgebra of a compact connected oriented surface with non-empty boundary. The smallest Torelli group of the surface in the sense of Putman is embedded into the kernel of the Turaev cobracket in the completed bialgebra. So a tensorial description of the completed bialgebra is desired. If the boundary of the surface is connected, a symplectic expansion introduced by Massuyeau relates the bialgebra to Kontsevich's "associative" Lie algebra. An approximative description of the Turaev cobracket in this case gives a geometric interpretation of the Morita traces on the cokernel of the Johnson homomorphism. On the other hand, if the genus of the surface is zero, a special expansion relates the bialgebra to the Lie algebra of special derivations on a free Lie algebra.
This talk is based on a joint work with Yusuke Kuno (Tsuda College).

Takahiro Kitayama
On an analogue of Culler-Shalen theory for higher dimensional representations (slides)

Culler and Shalen established a way to construct incompressible surfaces in a 3-manifold from ideal points of the SL2-character variety. We present an analogous theory to construct certain kinds of branched surfaces from limit points of the SLn-character variety. Such a branched surface induces a nontrivial presentation of the fundamental group as a 2-dimensional complex of groups. This is a joint work with Takashi Hara (Osaka University).

Cyril Lecuire
Extending mapping classes from the boundary of a handelbody (slides)

Consider a diffeomorphism f of a closed surface S and assume that S is the boundary of a handelbody. We want to know to which extent f is the restriction of a diffeomorphism of H.  This leads us to define the biggest subcompression body of H to which f extends. We explain how one can find out from the invariants sets of f whether this compression body is trivial or not. This is a joint work with Ian Biringer.

Gwénaël Massuyeau
Splitting formulas for the LMO invariant (slides)

For rational homology 3-spheres, there are two universal finite-type invariants: the Le–Murakami–Ohtsuki invariant and the Kontsevich–Kuperberg–Thurston invariant. These invariants take values in the same space of “Jacobi diagrams”, but it is not known whether they are equal. In 2004, Lescop found some relations satisfied by the variations of the KKT invariant when one replaces embedded rational homology handlebodies by others in a “Lagrangian-preserving” way. In this talk, we shall review the LMO invariant and show that it satisfies exactly the same relations. Our proof is an application of the LMO functor, which is a kind of TQFT extending the LMO invariant.

Greg McShane

Dynamics of the mapping class group (slides)

Let G be a Lie group. The mapping class group of a surface S has a natural action on the representation space of the fundamental group into G.
We discuss the nature of the action (ergodic, proper discontinous) for different G (compact, SL(n,R)) and for surface with boundary and cone points.

Ken'ichi Ohshika
Subgroups of mapping class groups generated by Dehn twists around meridians on splitting surfaces (slides)

We consider a splitting surface S of a Heegaard decomposition or a bridge decomposition. We are interested in a subgroup of the mapping class group of S generated by Dehn twists around "meridians" on S. We shall show that this group has a natural free product decomposition, and that there is an open subset in the projective lamination space of S on which this group acts locally properly discontinuously and freely. This is joint work with Makoto Sakuma and partly with Brian Bowditch.

Joan Porti

Varieties of representations in SL(n,C) (slides)

Given a three manifold, we consider its variety or representations in SL(n,C) and its variety of characters.
We discuss its local structure for representations of geometric origin, for instance the representation
induced by the holonomy of a hyperbolic manifold, or representations of Seifert manifolds.
We will also discuss explicit examples.

Leonid Potyagailo
Quasiconvexity in relatively hyperbolic groups and its applications (slides)

This is a joint work with V. Gerasimov (Belo Horisonte). An action of a group G on a compactum X is called relatively hyperbolic (RH-action) if it is discontinuous on the space of distinct triples Θ3 X and cocompact on the space of distinct couples Θ2 X. If G is finitely generated (or even countable) this definition is equivalent to the well-known Gromov-Farb-Bowditch definitions of the relative hyperbolicity (geometrical finiteness).
We provide several criteria for a subgroup H of G to be quasiconvex in a strong (absolute) and weak (relative) sense. As an application we obtain a stronger form of a recent theorem of Y. Matsuda, S. Oguni and S. Yamagata providing the existence of an equivariant homeomorphism between two RH-actions of G on different compacta.

Masatoshi Sato
On stable commutator length in the mapping class groups of punctured spheres (slides)

We give new upper and lower bounds on stable commutator length of some elements including half Dehn twists in the mapping class groups of punctured spheres. In particular, the lower bounds are obtained by calculating omega-signatures in Gambaudo-Ghys' paper. This talk is based on a joint work with Danny Calegari and Naoyuki Monden.

Yasushi Yamashita
The diagonal slice of SL(2,C)-character variety of free group of rank two (slides)

Let F2 be the free group of rank two. It is well known that SL(2,C)-character variety of F2 can be identified with C3 by the identification
ρ (trρ(A), trρ(B), trρ(AB)), where ρ is a representation of F2 into SL(2,C). In this talk, we study "the diagonal slice", that is, the subset of the character variety where trρ(A)=trρ(B)=trρ(AB). We (computationally) compare the "Bowditch set" and the discreteness locus. We also compute "rays" in the diagonal slice, and discuss some properties.