On the character variety of the three-holed projective plane

We study the (relative) SL(2,C) character varieties of the three-holed projective plane and the action of the mapping class group on them. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters. We consider the relationship with the previous work of the authors and S. P. Tan on the character variety of the four-holed sphere.Comment: 27 page

The asymptotic directions of pleating rays in the Maskit embedding

This article was born as a generalisation of the analysis made by Series, where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms' Relationship proved by the author and Series to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface S as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface S is the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to S, while the `bottom' end consists of triply punctured spheres, the remains of S when the pants curves have been pinched. Given a projective measured lamination l on S, the pleating ray P is the set of groups in M for which the bending measure of the top component of the boundary of the convex core of the associated 3-manifold is in the projective class of l.Comment: 29 pages, 5 figures, v2 fixes a typos in author's name in v

Higher signature Delaunay decompositions

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.Comment: 25 pages, 6 figure

Slices of quasifuchsian space

In Chapter 1 we present the background material about curves on surfaces. In particular we define the Dehn-Thurston coordinates for the set S = S(Σ) of free homotopy class of multicurves on the surface Σ. We also prove new results, like the precise relationship between Penner's and D. Thurston's definition of the twist coordinate and the formula for calculating the Thurston's symplectic form using Dehn-Thurston coordinates. For Chapter 2, let Σ be a surface of negative Euler characteristic together with a pants decomposition PC. Kra's plumbing construction endows Σ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the ith pants curve is denied by a complex parameter μi ∈ C. The associated holonomy representation ρμ : π1 (Σ)--> PSL(2;C) gives a projective structure on Σ which depends holomorphically on the μi. In particular, the traces of all elements ρ μ (γ), where γ ∈ π1 (Σ), are polynomials in the μi. Generalising results proved in [24; 40] for the once and twice punctured torus respectively, we prove in Chapter 2 a formula giving a simple linear relationship between the coefficients of the top terms of Tr ρμ (λ ), as polynomials in the μi, and the Dehn-Thurston coordinates of relative to PC. We call this formula the Top Terms' Relationship. In Chapter 3, applying the Top Terms' Relationship, we determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface Σ as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface Σ is the space of geometrically finite groups on the boundary of Quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of triply punctured spheres, the remains of Σ when the pants curves have been pinched. Given a projective measured lamination [η] on Σ, the pleating ray P = P[η] is the set of groups in M for which the bending measure pl+(G) of the top component ∂C+ of the boundary of the convex core of the associated 3-manifold H3=G is in the class [η]

Quasicircles and width of Jordan curves in CP1

We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature K∈[−1,0) and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes

The geometry of quasi-Hitchin symplectic Anosov representations

In this talk we will discuss quasi-Hitchin representations in $\mathrm{Sp}(4,\mathbb{C})$, which are deformations of Fuchsian (and Hitchin) representations which remain Anosov. These representations acts on the space $\mathrm{Lag}(\mathbb{C}^4)$ of complex lagrangian grassmanian subspaces of $\mathbb{C}^4$. This theory generalises the classical and important theory of quasi-Fuchsian representations and their action on the Riemann sphere $\mathbb{C} P^1 = \mathrm{Lag} (\mathbb{C}^2)$. In the talk, after reviewing the classical theory, we will define Anosov and quasi-Hitchin representations and we will discuss their geometry. In particular, we show that the quotient of the domain of discontinuity for this action is a fiber bundle over the surface and we will describe the fiber. The projection map comes from an interesting parametrization of $\mathrm{Lag}(\mathbb{C}^4)$ as the space of regular ideal hyperbolic tetrahedra and their degenerations. (This is joint work with D.Alessandrini and A.Wienhard.)Non UBCUnreviewedAuthor affiliation: University of VirginiaResearche