Let $ G $ be a group, let $ \varphi $ be an isomorphism of $ G $ onto a subgroup $ K $ of $ G $ , andlet $ G^{*} $ be a descending HNN-extension of $ G $ corresponding to $ \varphi $ .The potency of $ G $ is not inherited by $ G^{*} $ even in the simplest case, when $ G $ is an infinite cyclic group. We prove that if $ G $ is a finitely generated torsion-free nilpotent group (a polycyclic group); then the index $ m=[G:K] $ of $ K $ in $ G $ is finite and $ G^{*} $ is $ \pi $ -potent(virtually $ \pi $ -potent), where $ \pi $ is the set of all primes greater than $ m $ . We also provesome generalizations of this assertion. Some of the results of this work on the potencyof descending HNN-extensions are analogs of the well-known theorems on the residualfiniteness of the HNN-extensions.
