  We discuss two distinct realizations of the diffeomorphism group for metric
gravity, which give rise to theories that are classically equivalent, but
quantum mechanically distinct. We renormalize them in $d=2+\epsilon$
dimensions, developing a new procedure for dimensional continuation of metric
theories and highlighting connections with the constructions that previously
appeared in the literature. Our hope is to frame candidates ultraviolet
completions of quantum gravity in $d>2$ and give some perturbative mean to
assess its existence in $d=4$, but also to speculate on some potential
obstructions in the continuation of such candidates to finite values of
$\epsilon$. Our results suggest, as a common theme shared by all realizations,
the presence of a conformal window in $d$ which seems to extend to values
higher than four.
