arXiv subjects

Francis Bonahon

Publications and source records attributed to Francis Bonahon.

26 records · Page 2Linked to original sources

arXiv2007

Local representations of the quantum Teichmuller space

We introduce a certain type of representations for the quantum Teichmuller space of a punctured surface, which we call local representations. We show that, up to finitely many choices, these purely algebraic representations are classified by classical geometric data. We also investigate the family of intertwining operators associated to such a representations. In particular, we use these intertwiners to construct a natural fiber bundle over the Teichmuller space and its quotient under the action of the mapping class group. This construction also offers a convenient framework to exhibit invariants of surface diffeomorphisms.

math.GT↗

arXiv2007

Representations of the quantum Teichmuller space, and invariants of surface diffeomorphisms

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.

math.GT↗

arXiv2005

The metric space of geodesic laminations on a surface II: small surfaces

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is -1 over the logarithm of the Hausdorff metric. The key ingredient is an estimate of the Hausdorff metric between two simple closed geodesics in terms of their respective slopes.

math.GT↗

arXiv2004

The metric space of geodesic laminations on a surface: I

We consider the space of geodesic laminations on a surface, endowed with the Hausdorff metric d_H and with a variation of this metric called the d_log metric. We compute and/or estimate the Hausdorff dimensions of these two metrics. We also relate these two metrics to another metric which is combinatorially defined in terms of train tracks.

math.GT↗

arXiv2002

Kleinian groups which are almost fuchsian

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.

math.DG↗

arXiv2002

Variation of the Liouville measure of a hyperbolic surface

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic foliation.

math.DS↗

arXiv1997

Variations of the boundary geometry of 3--dimensional hyperbolic convex cores

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its bending is described by a measured geodesic lamination. We show that, as the hyperbolic metric of the 3--manifold M varies differentiably, the hyperbolic metric of the surface S varies in a C^1, but usually not C^2, manner. Differentiability properties for the bending measured lamination are conceptually less simple, because the space ML(S) of measured geodesic laminations on S has no natural differentiable structure. However, ML(S) is a piecewise linear manifold, and we also show that the bending measured geodesic lamination varies differentiably in a piecewise linear sense. The two results are proved simultaneously, mixing the differentiable and piecewise linear contexts. In particular, the 'corners' of the piecewise linear structure of ML(S) account for the fact that the metric of S is not a C^2 function of the metric of M. A by-product of the proof is a similar differentiablity property for Thurston's parametrization of complex projective structures on a surface by hyperbolic metrics and measured laminations.

dg-ga↗

arXiv1997

A Schlafli-type formula for convex cores of hyperbolic 3-manifolds

In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite hyperbolic 3--manifold M, as we vary the hyperbolic metric of M. In this case, the pleating locus of the boundary of the convex core is not constant any more, but we showed in an earlier paper that the variation of the bending of the boundary of the convex core is described by a geodesic lamination with a certain transverse distribution. We prove that the variation of the volume of the convex core is then equal to 1/2 the length of this transverse distribution.

dg-ga↗