Introduction

The first geometric invariant of groups was introduced by R. Bieri and R. Strebel for the special class of finitely generated metabelian groups [20]. Their work was motivated by an earlier result that every finitely presented soluble group with infinite cyclic quotient is an ascending HNN-extension with stable letter corresponding to a generator of the cyclic quotient and a finitely generated base [22]. It turned out that the new geometric invariant classifies the finitely presented groups in the class of all finitely generated metabelian groups [20]. Later on the definition of Bieri-Strebel was generalised for any finitely generated discrete group [18] and higher dimensional homological and homotopical analogues of this invariant were introduced by R. Bieri and B. Renz [19], [64]. We will discuss the precise definitions of these invariants in the following section. They are important as they determine the homological and homotopical types FPm and Fm of subgroups containing the derived subgroup.

In general the geometric invariants are very hard to compute and there are very few cases when they are calculated. One of them is the class of right angled Artin groups [48] which gave the first known examples of groups of type F P∞ which are not finitely presented [7]. Even in the class of metabelian groups the structure of the geometric invariants is not completely understood. There are two open conjectures in the metabelian case: the FPm-Conjecture and the Σm-Conjecture.