  We consider the Neumann Laplacian acting on square-integrable functions on a
triangle in the hyperbolic plane that has one cusp. We show that the generic
such triangle has no eigenvalues embedded in its continuous spectrum. To prove
this result we study the behavior of the real-analytic eigenvalue branches of a
degenerating family of triangles. In particular, we use a careful analysis of
spectral projections near the crossings of these eigenvalue branches with the
eigenvalue branches of a model operator.
