# Algorithmic Borel Finiteness?

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of them, given $V?$ Also, if the number of such manifolds is $f(V)$ are there any bounds on $f?$

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I think you're looking for this reference: front.math.ucdavis.edu/0811.2482 –  Ian Agol Feb 6 '12 at 17:49
I agree that this gives the bound, but I am not sure about the algorithmic aspect. I will try to investigate... –  Igor Rivin Feb 6 '12 at 19:55

There should be an algorithm in principle. There's a couple of approaches. Given the estimates in this paper, one can bound from above the degree of the invariant trace field of an arithmetic hyperbolic 3-manifold with volume $\leq V$. This in turn leads to a lower bound on the injectivity radius $\epsilon(V)$ ( it is conjectured that there is a universal lower bound to the injectivity radius of closed arithmetic 3-manifolds; this is true if one restricts to arithmetic manifolds defined over a number field of bounded degree by Lemma 4.9 and the fact that the Mahler measure of an integral polynomial of bounded degree is bounded). Now construct all manifolds of volume $\leq V$ with injectivity radius $\geq \epsilon(V)$. All arithmetic manifolds of volume $\leq V$ will appear among this list. One may perform this construction by bounding the number of tetrahedra in a triangulation (see e.g. Breslin), then gluing tetrahedra together in all possible ways, and computing whether they are arithmetic e.g. via Snap.