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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: – 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.

Another approach would be to construct all quaternion algebras over number fields of bounded degree with the appropriate ramification data coming from Borel's volume formula, and maximal orders in the quaternion algebras. Then compute presentations of the groups of units of the orders by applying Riley's algorithm to find a fundamental domain, then compute all finite index subgroups of bounded order by finding permutation representations.

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I know that this question hasn't seen any activity for several years, but I just now stumbled upon it and thought that it was worth mentioning that the second approach described by Ian Agol has a fairly long history and has been used in many contexts. For instance it was the mechanism by which Chinburg and Friedman identified the smallest hyperbolic 3-orbifold ( ). The same idea also underlies Prasad and Yeung's classification of fake projective planes ( – Ben Linowitz Sep 12 '14 at 17:29
The only issue is that if V is too large then it is not computationally practical to enumerate all number fields of bounded degree and discriminant. For instance we do not currently have a complete list of all totally real number fields of degree 10 and discriminant less than 15^10. So it would probably be hopeless to try to enumerate all arithmetic hyperbolic surfaces (their invariant trace fields are totally real) with volume less than V unless V were particularly small. – Ben Linowitz Sep 12 '14 at 17:32

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