Pregroups and the big powers condition

We study groups having the big powers property BP. It is proved that if a pregroup satisfies some natural axioms, then its universal group has this property. In particular, fundamental groups of some graphs of groups have the big powers property if BP holds for edge and vertex subgroups and a number of natural conditions are satisfied. The results obtained are applied to Lyndon's completions $U(P)^{\mathbb Z[t]}$ of the universal group $U(P)$ with $P$ satisfying the conditions mentioned.