The grafting map of Teichmüller space

Abstract

IntroductionOne of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients ofĈ by these groups reflect properties of the three-dimensional hyperbolic geometry of the quotients of H 3 by these groups. Yet, even though it has been over thirty-five years since Lipman Bers exhibited a holomorphic embedding of the Teichmüller space of Riemann surfaces in terms of the projective geometry of a Teichmüller space of quasi-Fuchsian manifolds, no corresponding embedding in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such an embedding. This embedding is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3-manifolds, we associate the bending measured lamination of the convex hull facing the fixed "conformal" end.The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33, p. 230 ). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture". This conjecture states that for a fixed measured lamination λ, the self-map of Teichmüller space induced by grafting a surface along λ is a homeomorphism of Teichmüller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application of this result that we give is to geometric coordinates on the Bers slice of QF , one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of Kleinian groups (see §5.2); thus we feel that this result is of interest in its own right.We now state our results and methods more precisely. Throughout, S will denote a fixed differentiable surface which is closed, orientable, and of genus g ≥ 2. Let T g be the corresponding Teichmüller space of marked conformal structures on S, and let P g denote the deformation space of (complex) projective structures on S (see §2 for definitions).

Circle packings on surfaces with projective structures and uniformization

“…The space ᏹᏸ g has a natural topology induced from the weak topology on measures on the set of undirected geodesics [Thurston 1979], and is known to be homeomorphic to euclidean space of dimension 6g − 6. A detailed account of the theory in terms of measured foliations (a notion equivalent to that of a measured geodesic lamination: see [Miller 1982]) can be found in [Fathi et al 1979;Thurston 1988].…”

Asymptoticity of grafting and Teichmüller rays

“…PSL 2 .R/ and thus has a canonical complex projective structure. Grafting, introduced by Thurston (see [22; 33], Scannell and Wolf [31] and Dumas and Wolf [11] for subsequent development) can be thought of as a way to deform the Fuchsian complex projective structure.…”

“…It is known that for any fixed lamination , the grafting map (X 7 ! gr X ) is a self-homeomorphism of T g [31].…”

“…is an approximating sequence of weighted simple closed curves (see [31]). The length in the Thurston metric on gr X of an arc intersecting , is its hyperbolic length on X plus its transverse measure.…”

Projective structures, grafting and measured laminations

“…A second, more synthetic geometric description of P.S/ is due to Thurston, and proceeds through the operation of grafting-a construction which traces its roots back at least to Klein [25,Section 50], with a modern history developed by many authors (Maskit [27], Hejhal [19], Sullivan-Thurston [41], Goldman [18], Gallo-KapovichMarden [15], Tanigawa [42], McMullen [29] and Scannell-Wolf [36]). The simplest example of grafting may be described as follows.…”