Hyperbolic Manifolds and Discrete Groups

485 Pages·2010·24.835 MB·English

by  Michael Kapovich (auth.)

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Hyperbolic Manifolds
and Discrete Groups
Michael Kapovich
Reprint of the 2001Edition
Birkha¨user
Boston • Basel • Berlin
Michael Kapovich
Department of Mathematics
University of California, Davis
1 Shields Ave.
Davis, CA 95616-8633
U.S.A.
[email protected]
Originally published in the Progress in Mathematics Series
ISBN978-0-8176-3904-4 (hardcover) e-ISBN978-0-8176-4913-5
ISBN978-0-8176-4912-8 (softcover)
DOI10.1007/978-0-8176-4913-5
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MathematicsSubjectClassification (2000): Primary: 30F40, 57M50; Secondary: 22E40, 53C20
(cid:2)c Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2001, First softcover printing 2009
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Michael Kapovich
Hyperbolic Manifolds
and Discrete Groups
Birkhauser
Boston • Basel • Berlin
Michael Kapovich
Department of Mathematics
University of Utah
Salt Lake City, UT 84109
U.S.A
Ubrary of Congress Cataloging-in-Publication Data
Kapovich, Michael, 1963-
Hyperbolic manifolds and discrete groups I Michael Kapovich.
p. em.-(Progress in mathematics; v. 183)
Includes bibliographical references and index.
ISBN 0-8176-3904-7-ISBN 3-7643-3904-7
1. Hyperbolic spaces. 2. Discrete groups. I. Title. II. Progress in mathematics (Boston,
Mass.); vol. 183.
QA685.K36 2000
516.9--{jc21 00-044529
CIP
AMS Subject Classifications: Primary: 30F40, 57M50; Secondary: 22£40, 53C20
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ISBN 0-8176-3904-7 SPIN 10535366
ISBN 3-7643-3904-7
Reformatted from the author's files by John Spiegelman, Philadelphia, PA.
Printed and bound by Hamilton Printing, Rensselaer, NY.
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9 8 7 6 5 4 3 2 1
Preface
The main goal of the book is to present a proof of the following.
Thurston's Hyperbolization Theorem ("The Big Monster"). Suppose that M is
a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the
interior of M admits a complete hyperbolic metric off inite volume.
This theorem establishes a strong link between the geometry and topology
of 3-manifolds and the algebra of discrete subgroups of Isom(JH[3). It completely
changed the landscape of 3-dimensional topology and theory of Kleinian groups.
Further, it allowed one to prove things that were beyond the reach of the standard
3-manifold technique as, for example, Smith's conjecture, residual finiteness of the
fundamental groups of Haken manifolds, etc. In this book we present a complete
proof of the Hyperbolization Theorem in the "generic case." Initially we planned
including a detailed proof in the remaining case of manifolds fibered over §1 as well.
However, since Otal's book [Ota96] (which treats the fiber bundle case) became
available, only a sketch of the proof in the fibered case will be given here.
The proof of the Hyperbolization Theorem is by induction on the steps of the
Haken decomposition of M along incompressible surfaces. Members of the Haken
hierarchy are manifolds M =No, N1, N2, N3, ... , N11, where each N; is obtained
by splitting N;-1 along a superincompressible surface and Nh is a disjoint union
of 3-balls. There are two cases:
(a) the "generic case," when the decomposition of M starts with an incom
pressible surface S that is not a virtual fiber and thus the manifold N1 is not an
interval bundle (or a disjoint union of two interval bundles) over a surface;
viii Preface
(b) the "exceptional case," when N1 is an interval bundle over a surface (or a
disjoint union of two such bundles). The most important example is when M fibers
over the circle with the fiberS, i.e., M is the mapping torus of a homeomorphism
s.
r: s~
Below is a sketch of the proof in the case (a) under the assumption that M
has empty boundary.
The first step of induction: each component of N, (which is a closed 3-ball)
admits a hyperbolic structure (just take a ball in the hyperbolic 3-space ). We skip
for a moment all the intermediate steps of the induction and consider the "last step
of induction" (the final gluing), which turns out to be the heart of the proof.
The final gluing. Assume that M is a closed atoroidal orientable Haken manifold
and that S c M is an incompressible surface that separates M into compact
components M1 , M2, each of which admits a hyperbolic metric and is not homotopy
equivalent to a surface (thus N1 = M1 u M2). LetS; := oM; and let r denote the
gluing mapping. We would like to find hyperbolic metrics g1, 82 on M1, M2 so
that we have the following:
The gluing map r is homotopic to an isometry f : Nbd(S1) ~ Nbd(S2)
between product neighborhoods of the surfaces S1, S2 that sends S1 to the surface
in oNbd(S2) that is different from S2.
Once this is done, we can glue the manifolds (M1, 81), (M2, 82) via the
isometry f and get a hyperbolic structure on the manifold M. Of course, there
will be an obstruction to the isometric gluing (i.e., to the existence of the metrics
81, 82 as above). The goal is to show that this obstruction is an incompressible
torus T c M, which thus corresponds to collections of disjoint cylinders in Mt
and M2. Therefore, if (say) the manifold M1 is acylindrical, then the isometric
gluing is unobstructed no matter what r is (since there are no incompressible tori).
Instead of the hyperbolic metrics we will try to find their holonomy representations
p; : n1 (M;) ~ P S L(2, C). The homomorphisms P1 and P2 have to be chosen to
form a commutative diagram:
n1 (Mt)
/ \.
nt(S) PSL(2,C)
\. /
Jrt (M2)
Once such p;s are found, they induce a homomorphism p : Jrt (M) ~ P SL(2, C)
(part (1) of the proof). To conclude that pis the holonomy of a hyperbolic struc
ture on M (i.e., that one can glue the corresponding metrics along neighborhoods
of S1, S2) one has to prove that P1, P2 satisfy some further conditions: a combi
nation theorem of Maskit (part (2) of the proof). Roughly speaking, these con
ditions require that the hyperbolic manifolds with boundary (M1, 8t), (M2, 82)
embed isometrically as deformation retracts in compact hyperbolic manifolds
(M~, 8~), (M~, 8~) that have convex boundary.
Preface ix
We now give more details. Let G; = rrt (M; ), i = 1, 2, G := (Gt, Gz).
Thurston's idea is to reduce the problem of finding the metrics g1, gz to the fixed
point problem for a certain map a of the Teichmiiller space T(G) = T(Gt) x
T(Gz) and then to prove the existence of a fixed point. As in many fixed point
theorems, one tries to find this fixed point as the limit of a sequence of iterations
a11(X) = [p11] E T(G). The spaces T(Gt), T(Gz) are spaces of hyperbolic
structures with convex boundary on the manifolds Mt, Mz: they are complete
locally compact metric spaces. The mapping a is a contraction:
d(a(p), a(q)) < d(p, q), unless p = q.
The key part of the proof of the existence of the fixed point is the Bounded Image
Theorem: it establishes relative compactness of the sequence [p in the Teichmiil
11]
ler space T(G). Algebraically, Teichmiiller spaces T(G;) correspond to equiv
alence classes of representations G; --+ P S L (2, C) that are induced by quasi
conformal homeomorphisms of the 2-sphere. The proof of precompactness breaks
into two parts: (1) the proof of existence of a pair of limiting representations
G; --+ P S L (2, C), (2) the proof of the fact that these representations are induced
by quasiconformal homeomorphisms.
Thurston's idea of the proof of (1) was based on a detailed study of the
geometry of pleated surfaces in hyperbolic manifolds, most of which was presented
by Thurston in the paper [Thu86a] and in the unpublished preprints [Thu87a],
[Thu87b ]. Instead of this approach we shall use a combination of geometry and
combinatorics: the theory of group actions on trees. The tree-theoretic approach to
proving precompactness of sequences of group representations was first developed
by Culler, Morgan and Shalen in the papers [CS83], [MS84], [Mor86], [MS88a,
MS88b]. The idea is to show that:
(i) Each "divergent" sequence of representations of G; corresponds to an
action of G; on a tree T; so that G; does not fix a point in T; (the "geometric part").
(ii) The action G; n. T; has to have a global fixed point (the "combinatorial
part").1
The "geometric" part of the proof in [CS83], [MS84], [Mor86] was actu
ally algebra-geometric; the geometric approach presented in this book is a version
of the geometric approaches of Bestvina [Bes88], Paulin [Pau89], and Chiswell
[Chi91 ]. The intuitive idea is that the ideal triangles in the hyperbolic space "ap
proximately look like" an infinite tripod, i.e., the union of three rays with the
common origin. If one divides the hyperbolic metrics by a very large constant,
then the "approximation" gets better. In the limit we get a tree.
The "combinatorial part" of Morgan-Shalen's proof [MS88a, MS88b] was
actually topological: it was based on an analysis of measured laminations in 3-
1W hat was actually proven in [MS88b J is that certain subgroups of the 3-manifold group have global
fixed points. To prove that the whole group fixes a point one has to apply a corollary of Skora's theorem
[Sko96).
x Preface
manifolds and has been replaced in this book by the more combinatorial Rips theory
ofg roup actions on trees; our discussion follows the paper of Bestvina and Feighn
[BF95]. As an alternative to this part of the proof, the reader can use either the
paper of Paulin [Pau97], which is essentially another version of the Rips Theory
(although many arguments are quite different from Rips' ideas) or the original
papers of Morgan and Shalen. The proof of Skora's theorem (needed in the proof
of part (1)) that we present in the book is again an application of the Rips Theory; our
discussion mainly follows Bestvina's paper [Bes97]. Very briefly, using the Rips
Theory we transform the action of G; on T; to an action of G; on a simplicial tree
where each edge stabilizer is cyclic and fixes a point in T;. Such action corresponds
to a decomposition of G; as an amalgamated free product (or an HNN-extension)
over a cyclic subgroup. This gives rise to an essential cylinder in M;. If M; is
acylindrical we get a contradiction. If M; is not acylindrical, then M; splits along
essential cylinders into submanifolds Z;j (the so-called JSJ decomposition) and by
applying the Rips Theory to each group rrt (Z;j) we conclude that it fixes a point in
T;. This is the most difficult part of the proof. On the other hand, the actions of G1
and Gz on the trees Tt and Tz are related: an element of rr1 (St) fixes a point in Tt
if and only if the corresponding (under the gluing map -r) element of rr1 (Sz) fixes
a point in Tz. Skora's theorem allows us to "collect" all the elements of rrt (S;) that
Si S2.
fix points in T; into subsurfaces Sf so that the gluing map -r carries to If
Sf -I= S;, then we get an incompressible torus in the manifold M by gluing cylinders
in M; whose boundaries are contained in as;. Thus both groups JTt(S;), i = 1, 2,
fix points in T;. Given this, one verifies that both groups G; fix points in the T; s,
which is a contradiction: this means that both sequences [p G; --+ P SL(2, IC)],
11 :
i = 1, 2, are relatively compact in
Hom(G;, PSL(2, C))/ PSL(2, C).
This concludes part (1) of the proof.
The proof of part (2) of the bounded image theorem presented here is probably
similar to the one that Thurston had in mind, although the details of this part of
the proof were not discussed in Thurston's preprints, and the corresponding part
of Morgan's outline is somewhat sketchy:
(a) First, one has to show that the representation (or rather a pair of repre
sentations) arising as the limit of Pn in part (1) does not have accidental parabolic
elements, i.e., nonparabolic elements of G; that are mapped to parabolic elements.
({3) Next, one has to show that the limit group is geometrically finite.
The proof of (a) is based on Sullivan's cusps finiteness theorem and the
theory of algebraic/geometric convergence of sequences of representations (de
veloped by Jorgensen, Thurston, and others). Briefly, each accidental parabolic
element corresponds to an essential annulus in M;; such annuli are glued by -r to
an incompressible torus in M, which is a contradiction.
Part ({3) is based on the theory of ends of hyperbolic 3-manifolds (developed
by Thurston, Bonahon, and others) and, again, algebraic/geometric convergence.