Take a look at Machlachlan and Reid's book "The Arithmetic of Hyperbolic 3-Manifolds".

Since finite volume hyperbolic structures are unique whenever an $n$-manifold ($n\geq 3$) has them, any invariants of the hyperbolic structure are invariants of the manifold. Hyperbolic manifolds are $K(\pi,1)$-spaces, so they're not just diffeo/homeomorphism invariants, but invariants of the homotopy-type.