The diffeomorphism invariance can be recovered in a weak-field perturbation about flat space, as was shown by Roček and Williams [181, 183], and there is some evidence for the existence of analogous zero-modes in perturbations of regular, non-flat tesselations, at least in 2d . Inspired by the perturbative analysis around flat space, Hamber and Williams  argue that a similar gauge invariance should persist even if one perturbs around an arbitrary non-flat background. They propose as a possible definition for such gauge transformations local variations of the link lengths that leave both the local volume and the local curvature terms invariant.
One may hope that in the non-perturbative Regge regime no gauge-fixing is necessary, since the contributions from zero-modes cancel out in the path-integral representation for operator averages [124, 122]. Menotti and Peirano [162, 164, 163, 161], following a strategy suggested by Jevicki and Ninomiya , have argued vigorously that the functional integral should contain a non-trivial Faddeev–Popov determinant. Their starting point is somewhat different from that adopted in the path-integral simulations (see also ). They treat piecewise flat spaces as special cases of differentiable manifolds (with singular metric), with the action of the full diffeomorphism group still well-defined. To arrive at a concrete representation for the Faddeev–Popov term which could be used in simulations seems at present out of reach.
Recently, Hamber and Williams [122, 123] have argued that the -lattice measure is the essentially unique local lattice measure over squared edge lengths (this is a special case of the one-parameter family of local measures of the form ; see also  for a related derivation). It does of course require a term with positive cosmological term in the action in order to suppress long edge lengths.
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