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
Chern Simons Theory and the volume of three-manifolds (slides)
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 or
will focus on the following
(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
(2) What kind of topological
information is enclosed in
the elements of vol(Ñ,
(3) Do these volumes satisfy the
covering property in the
sense of Thurston?
This is joint work
with Shicheng Wang.
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.
Unified quantum invariants of integral
homology 3-spheres associated to simple Lie algebras (slides)
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.
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
On the topology of the space of Kleinian once-punctured torus
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
We next explain Bromberg's theory
from the viewpoints of the trace coordinate and the complex
Especially, the shape of linear
slices which are close to the Maskit slice will be discussed.
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).
On an analogue of Culler-Shalen theory for higher dimensional
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
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.
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.
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.
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.
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
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.
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
The diagonal slice of SL(2,C)-character
variety of free group of rank two (slides)
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
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.