Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces.

*(English)*Zbl 0612.57010
Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham/Engl. 1984, Lond. Math. Soc. Lect. Note Ser. 111, 113-253 (1987).

[For the entire collection see Zbl 0601.00008.]

The authors give a careful and detailed exposition of important topics in Thurston’s theory of hyperbolic 3-manifolds. Let H denote hyperbolic 3- space, S its 2-sphere at infinity, L a closed subset of S not contained in a round circle, and C the convex hull of L in H. Given a simply connected component 0 of \(\partial C\) in H, there is a component 0’ of S- L which faces 0. Thurston’s theory of hyperbolic 3-manifolds requires an analysis of 0 and the relationship of 0 to 0’. A major component of the analysis is a theorem of D. Sullivan which says that there is a K- bilipschitz map between 0 with its path metric and 0’ with its Poincaré metric extending to the identity map on the common pointset boundary of 0 and 0’, where K is a universal constant independent of L.

The authors justify in detail all of the assertions of the Thurston- Sullivan theory with the exception of Sullivan’s very compelling contention that K may be taken equal to 2; their analysis succeeds only in showing that \(K\leq 67\). The paper is a boon to those mathematicians (probably the majority) who have difficulty filling in the omitted details in the Thurston and Sullivan proof sketches.

The authors give a careful and detailed exposition of important topics in Thurston’s theory of hyperbolic 3-manifolds. Let H denote hyperbolic 3- space, S its 2-sphere at infinity, L a closed subset of S not contained in a round circle, and C the convex hull of L in H. Given a simply connected component 0 of \(\partial C\) in H, there is a component 0’ of S- L which faces 0. Thurston’s theory of hyperbolic 3-manifolds requires an analysis of 0 and the relationship of 0 to 0’. A major component of the analysis is a theorem of D. Sullivan which says that there is a K- bilipschitz map between 0 with its path metric and 0’ with its Poincaré metric extending to the identity map on the common pointset boundary of 0 and 0’, where K is a universal constant independent of L.

The authors justify in detail all of the assertions of the Thurston- Sullivan theory with the exception of Sullivan’s very compelling contention that K may be taken equal to 2; their analysis succeeds only in showing that \(K\leq 67\). The paper is a boon to those mathematicians (probably the majority) who have difficulty filling in the omitted details in the Thurston and Sullivan proof sketches.

Reviewer: J.W.Cannon

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

30C62 | Quasiconformal mappings in the complex plane |