  We investigate off-shell perturbative renormalisation of pure quantum gravity
for both background metric and quantum fluctuations. We show that at each new
loop order, the divergences that do not vanish on-shell are constructed from
only the total metric, whilst those that vanish on-shell are renormalised by
canonical transformations involving the quantum fields. Purely background
metric divergences do not separately appear, and the background metric does not
get renormalised. We highlight that renormalisation group identities play a
crucial role ensuring consistency in the renormalisation of BRST
transformations beyond one loop order. We verify these assertions by computing
leading off-shell divergences to two loops, exploiting off-shell BRST
invariance and the renormalisation group equations. Although some divergences
can be absorbed by field redefinitions, we explain why this does not lead to
finite beta-functions for the corresponding field.
