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The fundamental group of a compact flat Lorentz space form is virtually polycyclic. (English) Zbl 0546.53039
The paper proves the assertion stated in the title. It is a general conjecture (perhaps first observed by L. Auslander, stated explicitly by J. Milnor [Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)] that a discrete group $$\Gamma$$ acting properly on $${\mathbb{R}}^ n$$ by affine transformations with compact quotient must be virtually polycyclic (equivalently contain a solvable subgroup of finite index).
The present paper affirmatively settles this conjecture in the case that $$\Gamma$$ preserves an inner product of signature (1,n-1). The techniques are heavily algebraic, using the classification of solvable subgroups of the orthogonal group O(1,n-1) and the fact that a discrete subgroup of O(1,n-1) has virtual cohomological dimension $$<n$$. These techniques are similar to those employed by D. Fried and W. M. Goldman [Adv. Math. 47, 1-49 (1983)] to prove Milnor’s conjecture when $$n=3$$. There is also some overlap of the results (when $$n=4)$$ with the recent I.H.E.S. preprint ”Complete flat spacetimes” by Fried.
These results are enhanced by the striking recent construction by Margulis of properly discontinuous groups acting by Lorentz isometries on $${\mathbb{R}}^ 3$$ which are not virtually solvable - in fact the groups $$\Gamma$$ constructed by Margulis are free groups of rank 2. These amazing examples show that Milnor’s conjecture above cannot be extended to the case of a noncompact quotient (Milnor’s original statement actually included the case of a noncompact quotient as well). Margulis’ paper appeared in Sov. Math., Dokl. 28, 435-439 (1983), translation from Dokl. Akad. Nauk SSSR 272, 785-788 (1983).

##### MSC:
 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 57S30 Discontinuous groups of transformations 22E25 Nilpotent and solvable Lie groups 53C30 Differential geometry of homogeneous manifolds
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