194 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDSGraph manifolds are generalizations of Seifert fibered manifolds and can bedefined as follows. A finite graph G = (V, E) is a finite collection V of vertices
and a collection E of edges joining pairs of vertices. If an edge e joins vertices u
and v, we say that e is incident to u and v. In this definition we allow for multiple
edges to be incident to the same pair of vertices and we also allow for an edge to
join a vertex to itself. The following is due to Waldhausen [427].
DEFINITION 31.54 (Graph manifold). A graph manifold M^3 is a closed 3-
manifold modeled on a finite graph G = (V, E), where to each vertex v E V there
corresponds a Seifert fibered manifold Mv whose boundary components are tori
and to each edge e E E t here corresponds a product manifold Me ~ T2 x [O, l]
such that if an edge e is incident to a vertex v, then there corresponds a gluing
of a boundary component of Mv to a boundary component of Me via a torus
diffeomorphism.
Based on the classification of Seifert fibered manifolds discussed above, graph
manifolds have been classified (see [427]). Graph manifolds are relevant to non-
singular solutions, which are discussed in the next few chapters, because of the
following result (see [58] and [59]).
THEOREM 31.55 (Cheeger and Gromov). Let M^3 be a compact 3-manifold
whose boundary components consist of 2-tori or whose boundary is empty. If M
admits a sequence of collapsing Riemannian metrics {gi}, i .e., metrics where(31.15) lim (max inj g; (x)
2
·max I Rm g; 1) = 0,
i-+oo xEM M
then M admits an F -structure^9 and hence is a graph manifold.6. Notes and commentary
For the proofs of most of the theorems stated in this chapter we refer the
reader to Thurston [401] and to Benedetti and P etronio [24] and the references
on p. 1 therein. See Ratcliffe [334] for a comprehensive and detailed treatment of
hyperbolic manifolds in any dimension.
§1. For a proof of the Cartan- Ambrose- Hicks theorem, see Theorem 1.37 in
Cheeger and Ebin [57].
See Chapter A of [ 24 ] or see Cannon, Floyd, Kenyon, and Parry [42] for detailed
discussions of the models of hyperbolic n-space.
For Lemma 31.5, see also Corollary A.3.8 in [24].
For the facts about Isom (IHin) in (31.3) and (31.4), see also Theorem A.4.1,
Theorem A .4.2, and Proposition A.5.13, all in [24].
For an exposition of the Brouwer fixed point theorem, see for example pp. 13 - 15
of Milnor [234].
For t he classification of isometries of IHin into elliptic, parabolic, and hyperbolic
types, see for example Propositions A.5.14 and A.5.18 in [24].
For a proof of Lemma 31.11, see Lemma D.2.4 in [24].
§2. For a proof of Theorem 31.14, see Theorem I.l and Corollary I.2, both in
Jaco [158].
For the Seifert and Van Kampen theorem, Theorem 31.22, see for example
p. 372 of Rolfsen [337].
(^9) See [58] for the definition of an F -structure.