Fundamental domains for affine Schottky groups preserving a quadratic form of signature (2n+2, 2n+1)

A conjecture due to Milnor asks whether any group of affine transformations acting properly discontinuously is virtually solvable. Margulis showed in 1983 that this was not the case; later, Abels, Margulis and Soifer constructed a family of counterexamples which are free groups preserving a quadratic form of signature $(d+1, d)$ (with $d$ odd). We shall give an explicit construction of a fundamental domain for these groups, which will allow us in particular to determine the topology of the quotient manifold.