Properties of the combinatorial Hantzsche-Wendt groups

The combinatorial Hantzsche-Wendt group Gn=〈x1,...,xn|xi−1xj2xixj2,∀i≠j〉 was defined by W. Craig and P.A. Linnell in [4]. For n=2 it is a fundamental of 3-dimensional oriented flat manifold with non cyclic holonomy group. We calculate the Hilbert-Poincaré series Gn,n≥1 Q F2 coefficients. Moreover, we prove that cohomological dimension Gn equal to n+1. Some other properties this are also considered.