Canonical Splitting of a Kleinian Group

摘要

If one starts with a Kleinian group of the second kind, G, where G is purelyloxodromic and geometrically finite, then the corresponding 3-manifold, M = (H(sup 3) union Omega(G))/G is compact and orientable, and so it can be split, first by disjoint properly embedded discs whose boundaries are homotopically distinct and non-trivial, and then using the characteristic submanifold theorem of Jaco-Shalen and Johannson. These splittings yield a finite set of essentially unique compact orientable 3-manifolds. There is also a characteristic submanifold theorem for orbifolds due to Bonahon and Siebenman. The results here are related; but the context is different and, within the class of hyperbolic 3-orbifolds, more general. The basic context is that of an analytically finite Kleinian group; that is, a discrete subgroup of PSL(2,C) with non-empty region of discontinuity that satisfies the conclusion of Ahlfors' finiteness theorem. The authors place no other restrictions on their group; that is, it may contain torsion, it may contain parabolic elements, and it may be infinitely generated. Rather than splitting in hyperbolic space, they split the group on the 2-sphere using systems of swirls and wheels--this is an invariant collection of simple closed curves and generalized simple closed curves. The authors also permit their splitting swirls and wheels to go through certain (doubly cusped) parabolic fixed points. Among other things, this means that their context is that of pinched Kleinian groups, where they enlarge the regular set to include doubly cusped rank 1 parabolic fixed points, with the obvious topology. In the geometrically finite case, this is equivalent to splitting a pared orbifold.