The fundamental group of a compact flat Lorentz space form is virtually polycyclic.

*(English)*Zbl 0546.53039The 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).

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 |