Notes on tameness

We give an account of the tameness theorem for complete hyperbolic 3-manifolds. Our presentation makes use of ideas from a number of different sources. We aim to give an account that is readily accessible from fairly standard geometrical and topological arguments, and adaptable to variable curvature. Let M be a complete hyperbolic 3-manifold with π1(M) finitely generated. Marden [Ma] asked if this implies thatM is topologically finite, i.e. homeomorphic (or equivalently diffeomorphic) to the interior a compact manifold with boundary. This was proven in the “indecomposable” case by Bonahon [Bon], and in general independently by Agol [Ag] and Calegari and Gabai [CalG]. Other accounts have since been given by Choi [Ch] and Soma [Som] (in the case without cusps). Our exposition is inspired by that of Soma, which gives another perspective on various constructions of [CalG]. The strategy we follow is broadly similar. However we phrase things differently, so as to remove the dependence on the end reduction theory [BrinT,My] and we include an independent argument for the relevant part of Souto’s result [Sou]. In this way, almost all our arguments, are built from first principles from results essentially known prior to Thurston’s work [T1]. Apart, that is, from one critical appeal to the hyperbolisation theorem for atoroidal Haken manifolds [T2,O,Ka], which is here used to deduce a purely topological statement (Theorem 1.7.1). There are a couple of places where the argument could be shortened a little by appeal to other more sophisticated results or machinery, as we point out in the relevant places. In [T1], Thurston formulated a notion of “geometrical tameness”, which was later shown to be equivalent to topological finiteness by Canary [Can1]. Since then, the term “tame” has commonly been regarded as synonymous with “topological finite”, though many of its consequences proceed via its geometrical interpretation. One consequence of tameness is the Ahlfors measure conjecture, which states that the limit set has either zero or full Lebesgue measure (see [Can1]). It is also a critical step in the classification of finitely generated kleinian groups. Other key ingredients in this are Thurston’s Ending Lamination Conjecture, [Mi,BrocCM] (see also [Bow2,Bow3]), as well as a description of which end invariants are realisable (see for example [KlS,NS]). Furthermore, in combination with various other results, it leads to a proof of the density conjecture of Bers, Sullivan and Thurston, namely that geometrically finite kleinian groups are dense among finitely generated ones. Arguments to deal with many (but not all) cases have been given by Bromberg and Brock [Brom,BrocB]. A key notion we use is that of a “polyhedral” surface, as in [Som]. The basic idea is that, appropriately interpreted, many of the essential constructions of “pleated surfaces”