Volumes of hyperbolic manifolds

Question

In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental?

Answer 1

First I'll throw out that the theorem you heard about, which is a corollary of the Mostow Rigidity Theorem, is more generally true for finite volume hyperbolic manifolds where it was proved by Prasad.

In the 2-dimensional case, by the Gauss-Bonnet theorem the volume equals $2 \pi$ times the Euler characteristic.

In dimension 3, the methods for evaluating volums are transcendental. Known exact formulas for these volumes are values of certain transcendental functions.

In the cusped case there is a commonly used method explained in this 1982 paper of Milnor, expressing volumes as linear combinations of values of the Lobachevskii function.

In the closed case, one uses Thurston's Dehn surgery theorem to relate volumes of closed hyperbolic 3-manaifolds with volumes of cusped hyperbolic 3-manifolds, as for example in this 1985 paper of Neumann and Zagier, or a later paper of Neumann, "Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds. Topology '90 (Columbus, OH, 1990), 243–271", which as the title suggests gives expressions for volume in terms of the Chern-Simons invariant.

I realize this does not address your exact question regarding rational, irrational, or transcendental, but I think the point of view might be worded as "Of course they're transcendental, but there's still a lot to be said". But having said that, I do not know the answer to this simple question: does there exist a finite volume hyperbolic 3-manifold with rational volume?