A new version of quantum gravity on discrete spaces (simplicial complexes)
is proposed. A theory of gravitation interacting with Dirac field is consid
ered. This theory is shown to be free of reparametrization anomaly. The pro
blem of axial gauge anomaly and the associated problem of the doubling of f
ermion states on a lattice are discussed. (C) 2001 MAIK "Nauka/Interperiodi
ca".