  We prove that residually finite mapping tori of polynomially growing
automorphisms of hyperbolic groups, groups hyperbolic relative to finitely many
virtually polycyclic groups, right-angled Artin groups (when the automorphism
is untwisted), and right-angled Coxeter groups have the cheap rebuilding
property of Abert, Bergeron, Fraczyk, and Gaboriau. In particular, their
torsion homology growth vanishes for every Farber sequence in every degree.
