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Elder siblings and the taming of hyperbolic 3-manifolds. (English) Zbl 0918.57004

Marden’s conjecture asserts that every hyperbolic 3-manifold \(M=\mathbb{H}^3/\Gamma\) with finitely generated fundamental group is tame, that is homeomorphic to the interior of a compact manifold with boundary. The conjecture is true for geometrically finite 3-manifolds (Marden) and for 3-manifolds with incompressible ends resp. a core with incompressible boundary (Thurston and Bonahon), but it remains open in the compressible case even in the simplest cases \(\pi_1M\cong\Gamma\cong\mathbb{Z}*\mathbb{Z}\).
In the present paper, a geometric criterion for tameness is given. As a motivation, the paper starts by describing what a wild Kleinian group \(\Gamma\), if it exists, should look like, by showing that any orbit of \(\Gamma\) would be knotted at arbitrarily large scales. This leads to the definition of the “elder sibling property” for a collection of open balls in hyperbolic 3-space \(\mathbb{H}^3\). By a Morse theory argument, it is shown that configurations of balls with the elder sibling property are unknotted. The main result of the paper states that if the \(\Gamma\)-orbit of a ball satisfies the elder sibling property, then \(M\) is tame. Also, a strict form of the elder sibling property holds for all convex cocompact Kleinian groups, and in fact characterizes such groups. “We do not expect the elder sibling property to hold for all hyperbolic 3-manifolds; rather, we hope it identifies a class of well-behaved manifolds that will serve as a point of departure for a deeper study of tameness”.

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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