GR Group Theory – Arbitrary distortion of cyclic subgroups of soluble groups

Accept $ G $ is a finally generated solvable group, and $ H $ an infinite cyclic subgroup of $ G $, Which possible functions can occur as subgroup distortion? $ H $ in the $ G $?

For polycyclic groups Osin and others have investigated this question. However, I am particularly interested in finitely generated, finite-rank solvable non-polycyclic groups. If this is already known, any references would be very happy.

Thank you for possible answers.