The Fundamental Group of a Compact Flat Lorentz Space Form Is Virtually Polycyclic

A flat Lorentz space form is a geodesically complete Lorentzian manifold of zero curvature. It is well known (see Auslander & Markus [3]) that such a space M may be represented as a quotient R/Γ, where R" is an ^-dimensional Minkowski space (n equals the dimension of M) and Γ is a group of Lorentz isometries acting properly discontinuously and freely on R. In particular the universal covering of M is isometric to R and the fundamental group ττλ{M) is isomorphic to Γ. Theorem. Let M be a compact flat Lorentz space form. Then πλ{M) is virtually poly cyclic. Recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups. This result affirms a conjecture of Milnor [13] in a special case. For discussion of this conjecture as well as another special case, we refer to Fried & Goldman [8]. The importance of this result is that it reduces the classification of compact flat Lorentz space forms to fairly elementary problems concerning Lie algebras and lattices in solvable Lie groups; we hope to pursue this classification in a future publication. For a description of this reduction and the classification in dimension 3, see Fried & Goldman [8]; in dimension 4 the classification is worked out in Fried [7]. One immediate consequence of the structure theory developed in Fried & Goldman [8, §1] and Kamishima [12] is the following Corollary. Let M be the compact flat Lorentz space form. Then M has a finite covering which is diffeomorphic to a solvmanifold. The outline of this paper is as follows. In the first section we collect some basic facts about the group E(n — 1,1) of isometries of ^-dimensional Minkowski space. In the second section we prove the theorem in the special