  We investigate some properties of geometric operators in canonical quantum
gravity in the connection approach \`a la Ashtekar, which are associated with
volume, area and length of spatial regions. We motivate the construction of
analogous discretized lattice quantities, compute various quantum commutators
of the type [area,volume], [area,length] and [volume,length], and find they are
generally non-vanishing.
  Although our calculations are performed mostly within a lattice-regularized
approach, some are -- for special, fixed spin-network configurations --
identical with corresponding continuum computations. Comparison with the
structure of the discretized theory leads us to conclude that anomalous
commutators may be a general feature of operators constructed along similar
lines within a continuum loop representation of quantum general relativity. --
The validity of the lattice approach remains unaffected.
