The BFK-gluing formula and relative determinants on manifolds with cusps

In this paper we use the BFK-gluing formula for zeta-determinants to express the relative determinant of Laplacians on a manifold with cusps by the product of the determinant of a Laplacian on a compact manifold with boundary augmented by the Dirichlet condition and the determinant of the Dirichlet-to-Neumann operator and some other ingredients. We use this result to compute some asymptotic expansions of zeta-determinants of Laplacians on a compact manifold with boundary augmented by the Dirichlet condition when we prolong the boundary part to cusps of infinite lengths.