5 Answering Igor's actual (?) question, linking to Hatcher.

[Edited several times] As the comments say, the answer to the first and hence to the second question is "no". Suppose that $M$ is the compact manifold and $N$ is its interior. Let $\rho$ be the given hyperbolic structure on $N$. If $M$ is without boundary then the volume of $\rho$ is finite and we are done.

Suppose instead that $M$ has boundary. Since $N$ is hyperbolic, via $\rho$, deduce $M$ is atoroidal (which includes aspherical). Thus $M$ is Haken. Place all tori in the boundary of $M$ into the paring locus $P$. By Thurston's hyperbolization theorem, the interior $N$ admits a hyperbolic metric, $\rho_0$, which is geometrically finite. (The convex core has finite volume and contains all torus boundary components.) See Theorem 1.43 in Kapovich's book.

[A brief note - your hypotheses can be weakened. You assumed (a) $N$ is the interior of a compact manifold and (b) N is hyperbolizable. This can be replaced by (a') $\pi_1(N)$ is finitely generated and the same (b). This is called the "tameness theorem", due to Agol and also Calegari-GabaiCalegari-Gabai.]

In the comments below (above?) Igor asks why an atoroidal manifold with torus boundary, and admitting an essential annulus, is Seifert fibered. This can be found as Lemma 1.16 on page 25 of Hatcher's three-manifold notes.

4 Cleaning up the logic.

[Edited] Edited several times] As the comments say, the answer to the first and hence to the second question is "no". Suppose that $M$ is the compact manifold and $N$ is its interior. Let $\rho$ be the given hyperbolic structure on $N$. If $M$ is without boundary then the volume of $\rho$ is finite and we are done.

Suppose instead that $M$ has boundary. Since $N$ is hyperbolic, via $\rho$, deduce $M$ is aspherical and atoroidal . Suppose that $M$ has boundary(which includes aspherical). Thus $M$ is Haken. Place all tori in the boundary of $M$ into the paring locus $P$. By Thurston's hyperbolization theorem, the interior $N$ admits a hyperbolic metric, $\rho_0$, which is geometrically finite. (The convex core has finite volume and contains all torus boundary components.) See Theorem 1.43 in Kapovich's book.

A brief note - your hypotheses can be weakened. You assumed (a) $N$ is the interior of a compact manifold and (b) N is hyperbolizable. This can be replaced by (a') $\pi_1(N)$ is finitely generated and the same (b). This is called the "tameness theorem", due to Agol and also Calegari-Gabai.

3 added 110 characters in body

[Edited] As the comments say, the answer to the first and hence to the second question is "no". Suppose that $M$ is the compact manifold and $N$ is its interior. Let $\rho$ be the given hyperbolic structure on $N$. If $M$ is without boundary , then the volume of $\rho$ is finite and we are done. Since $N$ is hyperbolic, via $\rho$, deduce $M$ is aspherical and atoroidal. Suppose that $M$ has boundary. Thus $M$ is Haken. Place all tori in the boundary of $M$ into the paring locus $P$. By Thurston's hyperbolization theorem, the interior $N$ admits a hyperbolic metric, $\rho_0$, where the which is geometrically finite. (The convex core is compact and of has finite volume and contains all torus boundary components.) See Theorem 1.43 in Kapovich's book. [I am looking for a precise reference for this!]

A brief note - your hypotheses can be weakened. You assumed (a) $N$ is the interior of a compact manifold and (b) N is hyperbolizable. This can be replaced by (a') $\pi_1(N)$ is finitely generated and the same (b). This is called the "tameness theorem", due to Agol and also Calegari-Gabai.

2 FIxed large mistake!
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