- MAXIMAL MODEL GEOMETRIES 7
Signature Lie group Description(-1, -1, -1) SU (2) compact, simple
(-1,-1,0) Isom ---------(JR.^2 ) solvable(-1, -1, +1) SL ---------(2, JR) noncompact, simple
(- 1, 0,0) nil nilpotent
(-1, 0, +1) sol solvable
(0, 0, 0) lREBlREBlR commutative
TABLE 1. 3-dimens10nal ummodular simply-connected Lie groupsgn is unimodular if its volume form is bi-invariant. This is equivalent to
the statement that the 1-parameter family of diff eomorphisms generated by
any left-invariant vector field preserves volume. Only unimodular Lie groups
admit compact quotients, because only those groups admit quotients of finite
volume. Thus one can make the following observation.
LEMMA 1.14. If (Mn,g) is a homogeneous model such thatlsom (Mn,g)
is diffeomorphic to Mn, then Mn is a Lie group. If moreover there exists a
discrete subgroup r of Isom (Mn, g) such that Mn /I' is compact, then Mn
is unimodular.
In [98], Milnor classified all 3-dimensional unimodular Lie groups. (His
method is sketched in Section 4 below.) Any simply-connected 3-dimensional
unimodular Lie group M^3 must be isomorphic to one of the six groups listed
in Table 1. Remarkably, the groups in this list make up five of the eight
maximal models that appear in the Geometrization Conjecture. (Thurston
discards Isom --------(JR.^2 ) because it is not maximal; see Remark 1.13.)
The key to classifying all maximal model geometries (M^3 , g, H) with
compact representatives is to look more closely at the isotropy group Q*. Let( M^3 , g, H) be a maximal model geometry, and let 9' denote the connected
component of the identity in g. Then 9' acts transitively on M^3 , so by
Lemma 1.12, all isotropy groups g~ of the 9' action are isomorphic. If g~
denotes the connected component of the identity in 9x, then the collection
9~/9~ forms a covering space of Mn. Since Mn is simply connected, this
covering must be trivial. It follows that all g~ are connected, hence isomor-
phic to a connected closed subgroup g~ of SO (3). Since the only proper
nontrivial Lie subalgebra of .so (3) is .so (2), one concludes that g~ is either
SO (3) itself or SO (2) or else the trivial group.
Following Thurston's argument in Chapter 3.8 of [123], one is thus able
to classify all 3-dimensional maximal model geometries which admit compact
representatives:
PROPOSITION 1.15. Let ( M^3 , g, g*) be a maximal model geometry repre-
sented by at least one compact 3-manifold. Then exactly one of the following
is true:
( 1) g* ~ SO ( 3), in which case either