Linear holonomy of Margulis space-times. (English) Zbl 0784.53040

Drumm, Todd A.

J. Differ. Geom. 38, No. 3, 679-690 (1993).

G. A. Margulis [Dokl. Akad. Nauk SSSR 272, 785-788 (1983; Zbl 0578.57012)] showed that there exist complete flat Lorentz space-times with free fundamental group, thereby disproving a conjecture of J. Milnor in [Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)]. The classification of the linear holonomy representation of these “Margulis space-times” is presented in this paper. In particular, any free discrete subgroup of \(SO(2,1)\) can occur as the image of the linear holonomy representation of a Margulis space-time. Surprisingly, the image of the linear holonomy representation can lie outside the identity component of \(SO(2,1)\) and contain parabolic elements of \(SO(2,1)\). The results of this paper are obtained by constructing fundamental domains for the actions of free subgroups of the affine group of affine space, which were first introduced by the author in an earlier paper.

Reviewer: Todd A.Drumm

MSC:

53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57S25	Groups acting on specific manifolds

Keywords:

Lorentz space-times; fundamental group