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.