Algorithmic Borel Finiteness? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:38:41Z http://mathoverflow.net/feeds/question/87680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87680/algorithmic-borel-finiteness Algorithmic Borel Finiteness? Igor Rivin 2012-02-06T15:58:45Z 2012-02-06T22:51:24Z <p>It is a theorem of Borel that there is a finite number of <em>arithmetic</em> 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 $f?$</p> http://mathoverflow.net/questions/87680/algorithmic-borel-finiteness/87733#87733 Answer by Agol for Algorithmic Borel Finiteness? Agol 2012-02-06T22:51:24Z 2012-02-06T22:51:24Z <p>There should be an algorithm in principle. There's a couple of approaches. Given the estimates in <a href="http://front.math.ucdavis.edu/0811.2482" rel="nofollow">this paper,</a> 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 <a href="http://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=1&amp;sqi=2&amp;ved=0CCIQFjAA&amp;url=http%253A%252F%252Fciteseerx.ist.psu.edu%252Fviewdoc%252Fdownload%253Fdoi%253D10.1.1.169.1318%2526rep%253Drep1%2526type%253Dpdf&amp;ei=61UwT-fYLK_YiQL56di7Aw&amp;usg=AFQjCNG_KrKCweWPxc93zOgUHTFIMLCZgg&amp;sig2=Wthc41Q1SERD8JX51hxrCA" rel="nofollow">Lemma 4.9</a> 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. <a href="http://front.math.ucdavis.edu/0711.0191" rel="nofollow">Breslin</a>), then gluing tetrahedra together in all possible ways, and computing whether they are arithmetic e.g. via <a href="http://www.ms.unimelb.edu.au/~snap/" rel="nofollow">Snap</a>. </p> <p>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. </p>