Hyperbolic 3-Manifolds – Volume with Toroidal Boundary

3-manifoldsgt.geometric-topologyhyperbolic-geometry

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from 3-Manifold Groups, page 18 (the link is editted) by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps).

Edits: The precise statement in my opinion should be:

Let $M$ be a compact three dimensional manifold with incompressible toroidal boundary (possibly none). If the interior of $M$ admits a hyperbolic structure, then $M$ either has finite volume, or is homeomorphic to $T^2\times I$.

Thanks for any solutions or hints.

Best Answer

I think you are trying to ask the following question.

Suppose that $M$ is a compact connected oriented three-manifold. Suppose that $M^\circ$, the interior of $M$, admits a hyperbolic metric. Then when must this hyperbolic metric have finite volume?

As Ryan points out, if $M$ is closed, then $M^\circ = M$ is compact and thus has finite volume. Also, as you note, the interior of $M = T^2 \times I$ admits (many) hyperbolic metrics, but all have infinite volume. There is another such manifold: namely the solid torus $D^2 \times S^1$.

I think that the place where you are confused (please correct me if I am wrong) is the case where $M$ has a boundary of higher genus. Here $M^\circ$ again always has infinite volume.

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