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This paper presents the construction of parametrices for the Gauss-Bonnet andHodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locallysymmetric 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 workof Melrose and Mazzeo. However, because they are not totally elliptic elementsin this calculus, it is not possible to construct parametrices for theseoperators within the $\phi$-calculus. We construct parametrices for them inthis paper using a combination of the $b$-pseudodifferential operator calculusof R. Melrose and the $\phi$-pseudodifferential operator calculus. Theconstruction simplifies and generalizes the construction done by Vaillant inhis thesis for the Dirac operator.  In addition, we study the mapping properties of these operators and determinethe appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplaceoperators are Fredholm. Finally, we establish regularity results for elementsof the kernels of these operators.

A Parametrix Construction for the Laplacian on Q-rank 1 Locally Symmetric Space