  The issue of local gauge invariance in the simplicial lattice formulation of
gravity is examined. We exhibit explicitly, both in the weak field expansion
about flat space, and subsequently for arbitrarily triangulated background
manifolds, the exact local gauge invariance of the gravitational action, which
includes in general both cosmological constant and curvature squared terms. We
show that the local invariance of the discrete action and the ensuing zero
modes correspond precisely to the diffeomorphism invariance in the continuum,
by carefully relating the fundamental variables in the discrete theory (the
edge lengths) to the induced metric components in the continuum. We discuss
mostly the two dimensional case, but argue that our results have general
validity. The previous analysis is then extended to the coupling with a scalar
field, and the invariance properties of the scalar field action under lattice
diffeomorphisms are exhibited. The construction of the lattice conformal gauge
is then described, as well as the separation of lattice metric perturbations
into orthogonal conformal and diffeomorphism part. The local gauge invariance
properties of the lattice action show that no Fadeev-Popov determinant is
required in the gravitational measure, unless lattice perturbation theory is
performed with a gauge-fixed action, such as the one arising in the lattice
analog of the conformal or harmonic gauges.
