Generation of polycyclic groups

Polycyclic groups are one of the best understood class of groups. For example most of the decision problems are decidable in this class, see [7]. It seems surprising therefore that it is still an open problem whether there exists an algorithm which finds d(G) for any polycyclic group G (given by say a set of generators and relations). This is unknown even in the case when G is virtually abelian. It is obvious that d(G) ≥ d(Ĝ) and when there is equality both the value of d(G) and a minimal generating set for G can indeed be found algorithmically. (Say by enumerating both the finite images and all possibilities for generating sets for G). In general d(G) − d(Ĝ) can be arbitrarily large even for metabelian groups G, see [5]. In fact Miklos Abert [1] has observed that there exist groups G with arbitrarily large d(G) while d(Ĝ) = 2. Fortunately for polycyclic groups the situation is not that bad. In [4] Linnell and Warhurst proved the following theorem using methods from commutative algebra and lattices over orders.