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. Let S be a surface of finite topological type, obtained by removing finitely many points from a compact surface without boundary, and with negative Euler characteristic. We consider complete hyperbolic metrics on the product S × ]−∞,∞[. The simplest ones are the fuchsian metrics defined as follows. Because of our hypothesis that the Euler characteristic of S is negative, S admits a finite area hyperbolic metric, for which S is isometric to the quotient of the hyperbolic plane H by a discrete group Γ of isometries. The group Γ uniquely extends to a group of isometries of the hyperbolic 3–space H respecting the transverse orientation of H ⊂ H, for which the quotient H/Γ has a natural identification with S×]−∞,∞[. A fuchsian metric is any metric on S× ]−∞,∞[ obtained in this way. Note that the image of H in H provides in this case a totally geodesic surface in S × ]−∞,∞[, isometric to the original metric on S. These examples can be perturbed to more complex hyperbolic metrics on S× ]−∞,∞[. See for instance [Th1][Mas]. A quasifuchsian metric on S× ]−∞,∞[ is one which is obtained by quasi-conformal deformation of a fuchsian metric. Equivalently, a quasifuchsian metric is a geometrically finite hyperbolic metric on S × ]−∞,∞[ whose cusps exactly correspond to the ends of S. These also correspond to the interior points in the space of all hyperbolic metrics on S × ]−∞,∞[ for which the ends of S are parabolic [Mar][Su]. If m is a quasifuchsian metric on S × ]−∞,∞[, the totally geodesic copy of S which occurred in the fuchsian case is replaced by the convex core C(m), defined as the smallest non-empty closed m–convex subset of S × ]−∞,∞[. If m is not fuchsian, C(m) is 3–dimensional and its boundary consists of two copies of S, each facing an end of S × ]−∞,∞[. The geometry of ∂C(m) was investigated by Thurston [Th1]; see also [EpM]. The component of ∂C(m) that faces the end S ×{+∞} is a pleated surface, totally geodesic almost everywhere, but bent along a family of simple geodesics; this bending is described and quantified by a measured geodesic lamination β(m) on S. Similarly, the bending of the negative component of ∂C(m), namely the one facing S×{−∞}, is determined by a measured geodesic lamination β(m). If Q(S) denotes the space of isotopy classes of quasifuchsian metrics on S × ]−∞,∞[ and if ML(S) is the space of measured geodesic laminations on S, the Date: February 1, 2008. This work was partially supported by grant DMS-0103511 from the National Science