TAILIEUCHUNG - Đề tài " Projective structures with degenerate holonomy and the Bers density conjecture "

We prove the Bers density conjecture for singly degenerate Kleinian surface groups without parabolics. 1. Introduction In this paper we address a conjecture of Bers about singly degenerate Kleinian groups. These are discrete subgroups of PSL2 C that exhibit some unusual behavior: • As groups of projective transformations of the Riemann sphere C they act properly discontinuously on a topological disk whose closure is all of C. • As groups of hyperbolic isometries their action on H3 is not convex cocompact. . | Annals of Mathematics Projective structures with degenerate holonomy and the Bers density conjecture By K. Bromberg Annals of Mathematics 166 2007 77 93 Projective structures with degenerate holonomy and the Bers density conjecture By K. Bromberg Abstract We prove the Bers density conjecture for singly degenerate Kleinian surface groups without parabolics. 1. Introduction In this paper we address a conjecture of Bers about singly degenerate Kleinian groups. These are discrete subgroups of PSL2C that exhibit some unusual behavior As groups of projective transformations of the Riemann sphere C they act properly discontinuously on a topological disk whose closure is all of C. As groups of hyperbolic isometries their action on H3 is not convex cocompact. Viewed as dynamical systems they are not structurally stable. These groups were first discovered by Bers Bers2 where he made the conjecture that will be the focus of our work here. Let M S X -1 1 be an I-bundle over a closed surface S of genus 1. We will be interested in the space AH S of all Kleinian groups isomorphic to 1 S . By a theorem of Bonahon this is equivalent to studying complete hyperbolic structures on the interior of M . A generic hyperbolic structure on M is quasi-fuchsian and the geometry is well understood outside of a compact set. In particular although the geometry of the surfaces S X t will grow exponentially as t limits to 1 or 1 the conformal structures will stabilize and limit to Riemann surfaces X and Y. Then M can be conformally compacti-fied by viewing X and Y as conformal structures on S X 1 and S X 1 This work was partially supported by grants from the NSF and the Clay Mathematics Institute. 78 K. BROMBERG respectively. Bers showed that X and Y parametrize the space QF S of all quasi-fuchsian structures. In other words QF S is isomorphic to T S X T S where T S is the Teichmuller space of marked conformal structures on S. Let the Bers slice BX be the slice of QF S obtained by fixing X and .

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