# Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper

Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.

W. Thurston states (and I quote)

A hyperbolic structure on the interior of a compact manifold $$M^3$$ has finite volume if and only if $$\partial M^3$$ consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.

His example 2.1 is simply the product of the 2-torus by the closed interval, $$M = \mathbb{T}^2\times I$$.

What did Thurston had in mind when he stated that? Why did he left the solid torus $$\mathbb{D}\times\mathbb{S}^1$$ out of this, say, classification? I ask that to understand if it was just a blunder or if he actually had a reason to exclude this case (which can be obviously obtained by the quotient $$\mathbb{H}^3/\{\tau\}$$, where $$\tau$$ is a parabolic translation in $$\mathbb{H}^3$$)?

• I think he just forgot his example. He also forgot few non-orientable examples. – Moishe Kohan Mar 14 '19 at 0:38
• I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it... – LLima Mar 14 '19 at 3:35
• I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary". – Lee Mosher Mar 14 '19 at 15:20

A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $$M^3$$ with incompressible boundary has finite volume if and only if $$\partial M^3$$ consists of tori, with the exception of $$T^2\times I$$. The latter admits no hyperbolic structure of finite volume.
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $$M^3$$ has finite volume if and only if $$\partial M^3$$ consists of tori and Klein bottles, with four exceptions which are $$T^2\times I, K^2\times I$$, $$D^2\times S^1$$ (the solid torus) and the total space of the unique nontrivial disk bundle over $$S^1$$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $$K^2$$ is the Klein bottle.