


SS15  Topologie de petite dimension et théorie géométrique des groupes / SS15  Low Dimensional Topology and Geometrical Group Theory Org: M. Boileau (Toulouse) et/and S. Boyer (UQAM)
 S. BASEILHAC, Institut Fourier, Grenoble I, 100 rue des maths, BP 74, 38402
St Martin d'Heres Cedex
State sum invariants of flat SL(2,C)connections
on 3manifolds and the volume conjectures

We have recently defined with R. Benedetti a new family of quantum
"dilogarithmic" invariants for compact oriented 3manifolds endowed
with a flat SL(2,C)connection. The quantum dilogarithmic
invariants are built via simplicial formula, based on triangulations
decorated with a heavy apparatus of combinatorial objects. These can
also serve to compute the volume and the ChernSimons invariant of
flat SL(2,C)connections. In this talk we present some strong
structural coincidences in the construction of both family of
invariants, and a volume conjecture relating the asymptotic behaviour
of the quantum dilogarithmic invariants to the volume and the
ChernSimons invariant of a non compact complete oriented hyperbolic
3manifold of finite volume.
 N. BERGERON, Université ParisSud, Bat. 425, Orsay 91405
Propriétés de Lefschetz pour les variétés
hyperboliques arithmétiques

De manière analogue aux théorèmes de Lefschetz pour les
variétés projectives, il semble exister des relations entre les
groupes de cohomologie d'une variété arithmétique et ceux de ses
sousvariétés totalement géodésiques. J'énoncerai, dans le
cas des variétés hyperboliques réelles et complexes, une
conjecture précise. J'essaierai ensuite de motiver cette conjecture,
d'en démontrer certains cas intéressant et d'expliquer ce qu'il
advient en rang supérieur.
 H. BODEN, McMaster University, 1280 Main St. W., Hamilton, ON L8S 4K1,
Canada
Calculations of the CassonCurtis SL(2,C)
invariant

This talk, which presents joint work with Cynthia Curtis, will focus
on the SL(2,C) analogue of Casson's invariant. This 3manifold
invariant was defined earlier by Curtis, and we give a simple closed
formula for the invariant for Seifertfibered homology 3spheres. One
way to establish this formula is to use the correspondence between the
SL(2,C) character varieties and the moduli spaces of parabolic Higgs
bundles of rank two. These results can then be utilized to provide
computations for families of 3manifolds arising as Dehn surgeries on
knots with Seifert slopes. For example, we describe computations of
the CassonCurtis for surgeries on twist knots. These computations
employ Curtis's surgery formula together with information about the
CullerShalen seminorms.
 F. BONAHON, Université de Californie Méridionale
Quantum hyperbolic geometry

We will discuss a connection between topological quantum field theory
and hyperbolic geometry. Following work of Chekhov, Fock and Kashaev,
we associate certain noncommutative algebras to manifolds of
dimensions 2 and 3. Our main result is that the representation
theory of these purely algebraic/combinatorial objects is essentially
controlled by the same data as hyperbolic metrics on the corresponding
manifolds. We use this construction to exhibit quantum invariants of
hyperbolic manifolds. This is joint work with Xiaobo Liu.
Nous établissons un lien entre la théorie topologique quantique
des champs et la géométrie hyperbolique. A partir de travaux de
Chekhov, Fock et Kashaev, nous associons des algèbres
noncommutatives aux variétés de dimension 2 et 3. Notre
résultat principal est que les représentations de ces objets
purement algébriques et combinatoires sont essentiellement
controllés par les mêmes données que les métriques
hyperboliques sur les variétés correspondantes. Nous utilisons
cette construction pour exhiber des invariants quantiques pour les
variétés hyperboliques. Ce travail a été effectué en
collaboration avec Xiaobo Liu.
 INNA BUMAGIN, McGill University, Montreal, Quebec, Canada
Isomorphism problem for fully residually free, or limit, groups

It was proved by O. Kharlampovich and A. Myasnikov that a JSJ
decomposition of a limit group can be constructed effectively. To
deduce solvability of the isomorphism problem, we show that every
automorphism of a limit group preserves an abelian JSJ decomposition
of the group (this part is adaptation of joint work with D. Wise).
Another ingredient of our proof is a generalization of a theorem
proved by Bestvina and by Paulin for hyperbolic groups: using Rips'
theory, we show that if G is a rigid vertex group in the JSJ
decomposition of a limit group, then G has a finite outer
automorphism group modulo edge subgroups. This is joint work with
O. Kharlampovich and A. Myasnikov.
 O. COLLIN, Université du Québec à Montréal
Cyclic surgeries, the Alexander polynomial and Floer Homology

The goal of this talk is to study which knots in S^{3} do not admit
nontrivial cyclic surgeries or cyclic surgeries with small
fundamental group. We will see that the Alexander polynomial contains
relevant information for this problem, once it is related to Floer
homology of knots. This is work in progress.
 T. FIEDLER, Université Paul Sabatier, 118 route de Narbonne, Toulouse
Simple knots can be detected with polynomial complexity of
degree four

We introduce a new technique in knot theory which comes from
singularity theory. There are two main applications: we give an
algorithm which detects the unknot, the trefoil, the figure eight
knot, ... with quartic complexity. We construct new calculable
knot invariants which are stronger than Vassiliev invariants.
 BENJAMIN KLAFF, Université du Québec à Montréal
Character varieties of hyperbolic 3manifolds

I'll discuss the birational equivalence between the following two
useful character varieties associated to a finitevolume hyperbolic
3manifold M:
(1) the "Dehn surgery" component X of the PSL(2,C)
character variety of M, and
(2) the image of X in the PSL(2,C) variety associated to
the boundary components of a compact core for M (under the
"restriction" map of characters induced by inclusion of the
boundary into M).
 M. LACKENBY, University of Oxford
A characterisation of large finitely presented groups

A group is known as "large" if it has a finite index subgroup that
admits a surjective homomorphism onto a nonabelian free group. Such
groups have many interesting properties, for example superexponential
subgroup growth. Possibly the strongest form of the virtually Haken
conjecture asserts that any hyperbolic 3manifold has large
fundamental group. This is known to be true in the cusped case. In my
talk, I will give a necessary and sufficient condition for a finitely
presented group to be large, in terms of the existence of a nested
sequence of finite index subgroups where successive quotients are
abelian groups with sufficiently large rank and order. The proof is
topological in nature, using a version of thin position for Cayley
graphs of finite groups.
 G. LEVITT, Université de Caen
JSJ splittings

JacoShalen and Johannson introduced canonical decompositions of
3manifolds, using tori and annuli. A similar theory has recently been
developed for arbitrary finitely presented groups. We describe basic
results and applications.
 S. TILLMANN, UQAM, 201 avenue du PrésidentKennedy, Montréal, QC
H2X 3Y7, Canada
Normal surfaces, angle structures and the character variety

I describe geometric splittings of cusped hyperbolic 3manifolds
associated to ideal points of the character variety. The techniques
used are geometric and combinatorial, involving normal surfaces, angle
structures and hyperbolic geometry.
 B. WIEST, Université de Rennes I
On the complexity of braids

We define a measure of "complexity" of a braid which is natural with
respect to both an algebraic and a geometric point of view.
Algebraically, we modify the standard notion of the length of a braid
by introducing generators D_{ij}, which are Garsidelike
halftwists involving strings i through j, and by counting powered
generators D_{ij}^{k} as log(k+1) instead of simply k.
The geometrical complexity is a certain natural measure of the amount
of distortion of the n times punctured disk caused by a
homeomorphism. Our main result is that the two notions of complexity
are comparable. The key rôle in the proofs is played by a technique
introduced by Agol, Hass, and Thurston. The methods of our proof,
combined with recent work by Hamenstädt, also yield a proof that
every braid has a s_{1}consistent representative of linearly
bounded length.
Joint with Ivan Dynnikov.

