  A cusp-decomposable manifold is a manifold constructed from a finite number
of complete, negatively curved, finite volume manifolds and identifying the
boundaries of truncated cusps by diffeomorphisms. Using properties of the
electric space of the universal cover of cusp-decomposable manifolds, we show
that the inclusion of walls and pieces induces quasi-isometric embeddings. We
also show that isomorphisms between fundamental groups of higher graph
manifolds preserve the decomposition into pieces.
