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Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups. (English) Zbl 1069.57004

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 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).

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 0718.30033
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