## On Thurston’s uniformization theorem for three-dimensional manifolds (chapter V).(English)Zbl 0599.57002

The Smith conjecture, Pure Appl. Math. 112, 37-126 (1984).
[For the entire collection see Zbl 0599.57001.]
This chapter deals with the hyperbolization of 3-manifolds. It presents a well-written and very readable outline of the structure of Thurston’s existence theorem for hyperbolic structures. But the reader should be aware from the beginning that also this outline is by far not complete and, as all other available outlines of the same subject, leaves out substantial parts of Thurston’s proof. This is partly due to the complexity of the very subject and partly due to the lack of written material which was available in those days. However, the article is very careful in isolating and precisely stating the steps which are covered. At crucial points the author refers to Thurston’s Princeton Notes and some other published sources which provide the reader with a considerable part of the necessary argument. Furthermore, these steps have been further clarified in the meantime by other authors, notably Morgan-Shalen and Bonahon, who view the subject from a different angle.
By the ”hyperbolization of 3-manifolds” is meant (since not completely settled yet) any effort of putting a ”hyperbolic structure” on a big class of 3-manifolds $$M^ 3$$, including the class of torus-free Haken 3- manifolds. Certainly, the first important step here is to clarify the very meaning of ”hyperbolic structures”. In any case, however, such a structure should imply the existence of a discrete and faithful representation $$\Pi_ 1M\to PSL_ 2 {\mathbb{C}}$$ and so the problem seems to be one in Kleinian group theory. This theory in turn is well developed and indeed crucial in any study of hyperbolic structures, but unfortunately has only few existence results, and the first major step towards existence was actually not provided by complex analysis but by geometry, carried out by E. M. Andreev in 1970 [Math. USSR, Sb. 10(1970), 413-440 (1971); translation from Mat. Sb., Nov. Ser. 81(123), 445-478 (1970; Zbl 0194.232)]. Indeed, Andreev’s paper gave the complete answer to the question when a 3-ball, given together with a ”boundary pattern”, can be realized as a hyperbolic polyhedron with totally geodesic faces (here a boundary pattern is a cell-decomposition of the boundary with an assignment of an angle to each edge). In essence it is shown that this realization is equivalent to the absence of essential triangles and squares in the 3-ball with boundary patterns.
The importance of Andreev’s result for 3-manifolds lies in the well-known fact that one arrives at such 3-balls with boundary patterns by splitting a Haken 3-manifold along a hierarchy of essential surfaces. Vice versa, one reobtains the original 3-manifold by glueing faces together successively, and therefore one might hope for a hyperbolic metric for $$M^ 3$$ in which the boundary $$\partial M^ 3$$ is totally geodesic. On the other hand any such ”totally geodesic” 3-manifold is covered by a $$\Pi_ 1M$$-equivariant region of $${\mathbb{H}}^ 3$$ whose frontier consists of planes. In this case the action of $$\Pi_ 1M$$ on the sphere at infinity of $${\mathbb{H}}^ 3$$ is especially nice: the regions of discontinuity of this action are simply interiors of circles.
In particular, any two such regions touch each other in at most one point, so $$M^ 3$$ itself has to be simple, i.e. admits no essential discs, annuli or tori, and consequently non-simple 3-manifolds cannot admit a totally geodesic structure. (Thurston conjectures that simple manifolds in turn have this property.) Thus the presence of annuli in $$M^ 3$$ again complicates the picture (it is known that annuli in $$M^ 3$$ are also responsible for exotic homotopy equivalences), and the only escape consists in weakening the naive notion of hyperbolic structure without losing to many of its features. The outcome is the crucial notion of a geometric finite structure, where a totally geodesic structure turns out to be geometric finite but not vice versa. This structure is characterized by various important properties (it is locally convex instead of totally geodesic, its fundamental region is finite sided, it is closed under finite coverings and the limit set of the action of $$\Pi_ 1M$$ on the sphere at infinity has Lebesgue measure zero). It is the existence of geometric finite structures for 3-manifolds which is the precise content of Thurston’s theorem.
In establishing this theorem one has to be aware that geometric finite structures are not rigid anymore. Fortunately however this is not too serious a setback since, according to a theorem of Ahlfors, Bers et al., the space of geometric finite structures is parametrized by the Teichmüller space, T($$\partial M)$$, of the boundary $$\partial M$$ (provided at least one such structure exists). This parametrization now is Thurston’s starting point for an induction construction of geometric finite structures (as a matter of fact this ”construction” is highly non- constructive). Since the induction beginning is essentially settled by Andreev’s theorem, it remains to solve the (much deeper) induction step, the ”glueing problem”.
The typical situation in this glueing problem is this: given a geometric finite 3-manifold M, together with a pair of disjoint incompressible surfaces $$S_ 1$$, $$S_ 2$$ in $$\partial M$$ and a glueing diffeomorphism $$\phi$$ : $$S_ 1\to S_ 2$$, the problem is to find a geometric finite structure for the quotient manifold M/$$\phi$$. Since, by hypothesis, $$\Pi_ 1M$$ is a Kleinian group, this is a very general form of what has long been known in Kleinian group theory as Klein’s combination problem. Thurston was able to demonstrate how to use 3-manifold topology to obtain the necessary numerical estimates needed for a solution of this problem. The core of the argument deals with the case $$\partial M=S_ 1\cup S_ 2$$, and sections 7-12 of the author’s article are devoted to this case.
Since the originally given geometric finite structure $$\sigma\in T(\partial M)$$ of M is not necessarily totally geodesic, it also might not glue together to form a similar structure for M/$$\phi$$. The idea is now to vary $$\sigma$$ in a systematic way to a structure with this very glueing property. Now, in a natural way (as explained in section 9), $$\phi$$ induces a homeomorphism $$\phi_*: T(\partial M)\to T(\partial M)$$ and any fixed point of this map satisfies the above glueing property. It is described in sections 8 and 9 how this easily follows from Maskit’s combination theorems. To really find such a fixed point and so to establish its existence, one studies the iterates of the original point $$\sigma$$ under $$\phi_*$$. It is shown (section 10) that $$\phi_*$$ is distance decreasing (w.r.t. the Teichmüller distance) and that the existence of the (necessarily unique) fixed point therefore follows from the boundedness of the sequence $$\phi^ n_* \sigma$$ (it appears as the limit of this sequence). This boundedness in turn is proven in two big steps, each one requiring specific and novel techniques.
In the first step the natural embedding of the space of geometric finite structures of M into the space AH(M) of discrete, faithful representations $$\Pi_ 1M\to PSL_ 2 {\mathbb{C}}$$, mod. conjugacy, is utilized. Indeed, this step consists in showing that the sequence $$\phi_*^ n \sigma$$ (or rather some sub-sequence) has an algebraic limit, i.e. converges in AH(M). For this it has to be shown that the lengths for curves in M stay bounded w.r.t. the metrics given by the points $$\phi^ n_* \sigma \in T(\partial M)$$, i.e. that the characters of conjugacy classes of elements of $$\Pi_ 1M$$ stay bounded. Given the characteristic submanifold V of M, two types of curves are here to be distinguished: those which can and those which canot be deformed into M- V. If $$V=\emptyset$$, i.e. if M is simple, it is known that the whole space AH(M) is compact. The author refers here to W. P. Thurston [Ann. Math., II. Ser. 124, 203-246 (1986)] which in turn is based on the analysis given in Thurston’s Princeton Notes. Since the publication of the book, the author and P. B. Shalen worked out a different approach to this deep result, using group actions on $${\mathbb{R}}$$-trees and analyzing laminations associated to the fixpoints of such actions [ibid. 120, 401-476 (1984; Zbl 0583.57005)].
At this point the reader is advised to consult this interesting paper too. Unfortunately, however, the available material covers only the absolute case, i.e. the case $$V=\emptyset$$. It still remains to show that curves in the frontier $$\partial V-\partial M$$ stay bounded (this is enough to show that all curves in M-V stay bounded). This relative case is formulated as theorem 11.2. in the author’s article, but to my knowledge no written proof of this result is available yet. Now, the boundedness of the curves in M-V forces (under the hypothesis M-V$$\neq \emptyset)$$ the boundedness for all curves in M. This is not true for all sequences in AH(M), but fortunately holds true for the sequence $$\phi^ n_* \sigma$$. An explicit argument for this fact (at least for all those curves in $$\partial M)$$ is presented in section 11 (here the argument is similar to the corresponding step in the study of exotic homotopy equivalences [the reviewer, Homotopy equivalences of 3-manifolds with boundaries (Lect. Notes Math. 761) (1979; Zbl 0412.57007)]). This in turn then completes the discussion in the case M-V$$\neq \emptyset.$$
Having established the existence of the algebraic limit for the sequence $$\phi^ n_* \sigma$$, it now remains to show that this limit is in fact geometric finite. This is the second step and is the content of section 12. At this point recall that for arbitrary sequences of Kleinian groups (and even quasi-Fuchsian groups) an algebraic limit need not be geometric finite. Indeed, according to the discovery of L. Bers [Ann. Math., II. Ser. 91, 570-600 (1970; Zbl 0197.060)], one always has to be aware not only of accidental parabolics (i.e. the occurrence of additional cusp-ends), but also of the degeneration of ends in general, so that the measures of the limit sets might suddenly increase. Now, in the case at hand, accidental parabolics can be excluded comparatively easily, which is described in full detail in section 12. The reason is that, because of the special nature of the sequence $$\phi^ n_* \sigma$$, accidental cusp-ends would come in pairs ”joined” by essential annuli in M. Moreover, these cusp-ends would fall into two systems corresponding under the glueing diffeomorphism, and so the above annuli would fit together to form essential tori in M/$$\phi$$, which is certainly impossible.
A different and much more serious matter is to exclude degenerations of ends. Here an analysis of degenerated Kleinian groups and their ends is required, a problem which has puzzled experts since Bers’ discovery of such groups. It has been completely solved by Thurston for limits of geometric finite groups. Parts of his analysis can be found in his Princeton Notes, and the article refers to these Notes at this stage. In the meantime F. Bonahon pushed Thurston’s ideas still further, injecting Ruelle-Sullivan’s idea of currents for the relevant study of limits of singular curves on surfaces (needed for the study of ends), and recently [ibid. 124, 71-158 (1986)] came up with a proof of (Thurston’s problem) the result that ends of all finitely generated Kleinian groups, different from free products, are always ”tame” and hence clarifying the picture greatly. In section 12 the reader will find the argument why ”tameness”, in the situation at hand, would imply that the algebraic limit of a degenerating sequence $$\phi^ n_* \sigma$$ represents a Stallings fibration and why this is impossible. Thus after all no accidental parabolics and no degenrations may occur, so the algebraic limit of $$\phi^ n_* \sigma$$ is geometric finite, i.e. a limit in the (sub)space of geometric finite structures of M, and so the required fixpoint of $$\phi_*$$ is therefore found.
After this achievement essentially two situations still remain open: (1) $$V=M$$ and (2) $$\partial M\neq S_ 1\cup S_ 2.$$
If $$V=M$$, then M/$$\phi$$ has to be a Stallings fibration (or doubly covered by such a fibration). This important special case is briefly touched in section 13, and is also the subject of a Bourbaki report given by D. Sullivan [Lect. Notes Math. 842, 196-214 (1981; Zbl 0459.57006)]. Details can also be found in the article by W. Thurston [”Hyperbolic structures on 3-manifolds. II: Surface groups and 3-manifolds which fibre over the circle (to appear)].
If $$\partial M\neq S_ 1\cup S_ 2$$, then boundary patterns play an important role again. The idea in this situation is to reduce this case to the case where $$S_ 1$$ and $$S_ 2$$ are closed surfaces by simply doubling M along $$\partial M-S_ 1\cup S_ 2$$. The obstruction for this idea however is again the fact that $$\partial M-S_ 1\cup S_ 2$$ need not be totally geodesic. But fortunately one can put a boundary pattern on $$\partial M$$ in such a way that all faces are totally geodesic. Now, instead of doubling along $$\partial M-S_ 1\cup S_ 2$$, one reobtains the required situation by successively doubling along all faces of the boundary pattern which are contained in $$\partial M-S_ 1\cup S_ 2$$. The formal argument (based on Thurston’s generalization of Andreev’s theorem as given in his Notes) is presented comprehensively in section 14. This then completes the author’s outline.
I should add that Thurston discovered seven more geometries for 3- manifolds (besides the hyperbolic structure) and showed that this is a complete list. An excellent reference for this topic is Scott’s report [P.Scott, Bull. Lond. Math. Soc. 15, 401-487 (1983; Zbl 0561.57001)]. From this discovery Thurston then developed his now famous conjecture that every compact 3-manifold has a canonical decoposition into pieces with geometric structures, a conjecture which many topologists believe to suggest a good global picture for 3-manifolds.
Reviewer: K.Johannson

### MSC:

 57-06 Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes 57N10 Topology of general $$3$$-manifolds (MSC2010) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 00Bxx Conference proceedings and collections of articles 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)