On torsion-free groups with finite regular file bases
  The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group $G$ has a finite system of generators $a_1$, ..., $a_n$ such that every element of $G$ has a unique presentation in the form $a_1^{k_1}... a_n^{k_n}$ where $k_i\in\mathbb Z$, is it true that $G$ is virtually polycyclic? The answer is ``not always.'' A counterexample is constructed in this paper as a group presented by generators and defining relations. 
Alexey Muranov