The linear part of an affine group acting properly discontinuously and leaving a quadratic form invariant

Abstract

In this paper we study the dynamics of properly discontinuous and crystallographic affine groups leaving a quadratic from of signature (p, q) invariant. The main results are: (I) If p − q ≥ 2, then the linear part of the group is not Zariski dense in the corresponding orthogonal group. (II) If q = 2 and the group is crystallographic, then the group is virtually solvable. This proves the Auslander conjecture for this case.