CHAPTER 2
Complete affine flows
The purpose of this chapter is to give a proof of Theorem 1 in Introduction.
This chapter is organized as follows. In Sect. 1, we show that a complete affine flow
(/?ona closed 3-manifold is virtually isomorphic to a flow in Example 1.4 provided p
is a Seifert fibration or admits a cross section. In most parts of later sections, we try
to find out a cross section of p. Sect. 2 is devoted to the case where the holonomy
group is contained in Nil. Thanks to the Mal'cev theorem on nilpotent Lie groups
and to the Waldhausen theorem on Haken 3-manifolds, we manage to show that a
flow with nilpotent holonomy is virtually isomorphic to a flow in Example 1.4. In
Sect. 3 and 4 we deal with general cases and reduce them to the results of Sect. 1
and 2. There is a dichotomy of flows according to whether it admits a periodic orbit
or not. In Sect. 3 flows with periodic orbits are treated and Sect. 4 is devoted to the
study of flows without periodic orbits. In both cases, flows are often accompanied
with codimension one foliations derived from linear foliations on R2, which makes
easier the investigation.
1. Theorem of Nagano-Yagi and its application
General affine structures on closed surfaces are classified by Nagano and Yagi
[32]. But for the purpose of this paper we pay attention only to complete affine
structures, that is, structures whose developing maps are homeomorphisms onto
R2.
Hence their theorem will be stated in a truncated form so as to cover this case
only.
There are two distinct simply transitive actions of
R2
on
R2.
That is, the
actions given by E\ and Ei of Introduction. Given a uniform lattice I\ of E^ one
obtains a complete affine structure on I\ \ Ei ~
T2,
called type Ei.
THEOREM
4. A complete affine structure on a closed oriented surface E is
either of type E\ or of type E2. In particular E is homeomorphic to T2.
Call a complete affine flow homogeneous (resp. virtually homogeneous) if it is
isomorphic (resp. virtually isomorphic) to a flow in Example 1.4. In this section, we
prepare two sufficient conditions 1.1 and 1.2 for a given affine flow to be virtually
homogeneous. Both are easy consequences of Theorem 4.
PROPOSITION
1.1. If a complete affine flow if on a closed 3-manifold M is a
Seifert fibration, then p is virtually homogeneous.
PROOF.
Passing to an appropriate finite covering if necessary, we may assume
that ip is an oriented
Sx-bundle
flow over an oriented surface E. Clearly the trans-
verse structure of p induces a complete affine structure on E. In particular E is
homeomorphic to
T2
and the fundamental group iri(M) is nilpotent. It is well
known, easy to show using Mal'cev's theorem [28], that an S1 bundle M T2 is
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