Properties of the combinatorial Hantzsche-Wendt groups

Abstract

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 group of 3-dimensional oriented flat manifold with non cyclic holonomy group. We calculate the Hilbert-Poincaré series of Gn,n≥1 with Q and F2 coefficients. Moreover, we prove that the cohomological dimension of Gn is equal to n+1. Some other properties of this group are also considered.