Normal Abelian subgroups and Euler characteristic

We generalize Rosset's theorem which states that the Euler characteristic of a group G of type FL_C vanishes if G contains a torsion free normal subgroup. In our case the subgroup is allowed to have torsion (but must also have elements of infinite order). Under similar conditions on the regular covering of a finite CW-complex X, it is shown that the Euler characteristic of X is 0; this includes the special case where X is nilpotent.