  This paper presents the construction of parametrices for the Gauss-Bonnet and
Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally
symmetric spaces. These operators are, up to a scalar factor,
$\phi$-differential operators, that is, they live in the generalised
$\phi$-calculus studied by the authors in a previous paper, which extends work
of Melrose and Mazzeo. However, because they are not totally elliptic elements
in this calculus, it is not possible to construct parametrices for these
operators within the $\phi$-calculus. We construct parametrices for them in
this paper using a combination of the $b$-pseudodifferential operator calculus
of R. Melrose and the $\phi$-pseudodifferential operator calculus. The
construction simplifies and generalizes the construction done by Vaillant in
his thesis for the Dirac operator.
  In addition, we study the mapping properties of these operators and determine
the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace
operators are Fredholm. Finally, we establish regularity results for elements
of the kernels of these operators.
