Joan Porti

Deformations of reducible representations of 3-manifold groups into PSL2(C)

Let M be a 3-manifold with torus boundary which is a rational homology circle. We study deformations of reducible representations of π1(M) into PSL2(C) associated to a simple zero of the twisted Alexander polynomial. We also describe the local structure of the representation and character varieties. AMS Classification 57M27; 20C99; 57M05

TWISTED COHOMOLOGY FOR HYPERBOLIC THREE MANIFOLDS

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using techniques of Matsushima-Murakami. We give some applications to local rigidity.

Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {Tn(M;η)}. Roughly speaking, Tn(M;η) is the Reidemeister torsion of M respect to the representation given by the composition of the lift of the holonomy representation defined by η, and the n-th dimensional irreducible complex representation of SL(2,C). In the present work, we focus on two aspects of this invariant: its asymptotic behaviour and its relationship with the complex length spectrum of the manifold. Concerning the former, we prove that for suitable spin structures, log|Tn(M;η)| ∼ −n 2 VolM

Deforming Euclidean cone 3–manifolds

Given a closed orientable Euclidean cone 3‐manifold C with cone angles and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles < . We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbolic ones and we also deduce global rigidity for Euclidean cone structures. 57M50

The boundary of the Gieseking tree in hyperbolic three-space

Abstract We give an elementary proof of the Cannon–Thurston Theorem in the case of the Gieseking manifold. We do not use Thurstons structure theory for Kleinian groups but simply calculate with two-by-two complex matrices. We work entirely on the boundary, using ends of trees, and obtain pictures of the regions which are successively filled in by the Peano curve of Cannon and Thurston.

Regenerating hyperbolic and spherical cone structures from Euclidean ones

Abstract We show that, in some cases, a Euclidean cone structure on a closed 3-manifold can be deformed into hyperbolic or spherical cone structures by moving the singular angle. We describe other deformations on the complement of the singular set by using generalized Dehn surgery parameters. In order to do that, we study the relationship between algebraic deformations of the holonomy representation and deformations of the geometric structure.

Uniformization of small 3-orbifolds

Abstract We prove a uniformization theorem for small compact orientable 3-orbifolds, that implies Thurstons orbifold theorem.

Mayberry-Murasugi’s formula for links in homology 3-spheres

We prove the Mayberry-Murasugi formula for links in homology 3-spheres, which was proved before only for links in the 3-sphere. Our proof uses Franz-Reidemeister torsions.

Infinitesimal projective rigidity under Dehn filling

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.

Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

For noncompact semisimple Lie groups G with finite center, we study the dynamics of the actions of their discrete subgroups Gamma < G on the associated partial flag manifolds G / P. Our study is based on the observation, already made in the deep work of Benoist, that they exhibit also in higher rank a certain form of convergence-type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the Gamma-action on various domains of proper discontinuity, in particular on domains in the full flag manifold G / B. In the regular case (eg of B-Anosov subgroups), we prove the nonemptiness of such domains if G has (locally) at least one noncompact simple factor not of the type A(1), B-2 or G(2) by showing the nonexistence of certain ball packings of the visual boundary.