Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups.
For a compact, hyperbolizable 3-manifold \(M\) whose non-empty boundary contains no tori, let \(CC_0(M)\) denote the space of convex cocompact uniformizations of \(M\) (which is a component of the interior of the space \(AH(\pi_1(M))\) of all marked hyperbolic 3-manifolds homotopy equivalent to \(M\)). The main result of the present paper states that if a hyperbolic 3-manifold in the boundary of \(CC_0(M)\) has empty conformal boundary (i.e., all of its ends are geometrically infinite), then it can be approximated by maximal cusps, that is by geometrically finite hyperbolic 3-manifolds such that every component of their conformal boundary is a thrice-punctured sphere. Since, according to Thurston's ending lamination conjecture, geometrically finite hyperbolic 3-manifolds correspond to the ``rational points'' in the boundary of \(CC_0(M)\), the main result can be thought of as asserting that, in general, the ``most irrational'' points in \(\partial CC_0(M)\) can be approximated by rational points. As a consequence, it is shown that maximal cusps are dense in \(\partial CC_0(M)\) if the boundary of \(M\) is connected; in particular, if \(M\) is a handlebody \(H_g\) of genus \(g \geq 2\) then maximal cusps are dense in the boundary of Schottky space \(CC_0(H_g)\). The paper makes use of the analytic machinary developed by \textit{C. McMullen} [Ann. Math. 133, 217--247 (1991; Zbl 0718.30033)] who was the first to study the density of ``rational points'' in the boundary of deformation spaces of Kleinian groups, showing that ``maximal cusps'' are dense in the boundary of any Bers slice of quasifuchsian space (i.e., fixing the conformal structure on one boundary component of a 3-manifold which is the product of a surface with an interval; in this situation, the conformal boundary of a maximal cusp consists of one copy of the surface and a collection of thrice-punctured spheres).
scientific article; zbMATH DE number 2171934