  We prove new vanishing results on the growth of higher torsion homologies for
suitable arithmetic lattices, Artin groups and mapping class groups. The growth
is understood along Farber sequences, in particular, along residual chains. For
principal congruence subgroups, we also obtain strong asymptotic bounds for the
torsion growth.
  As a central tool, we introduce a quantitative homotopical method called
effective rebuilding. This constructs small classifying spaces of finite index
subgroups, at the same time controlling the complexity of the homotopy. The
method easily applies to free abelian groups and then extends recursively to a
wide class of residually finite groups.
