In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used the perspective of flat structures to get the same results [3] (and apparently corrected the Deligne–Mostow list by slightly extending it, according to another paper of Mostow). In the Thurston perspective, the idea is to enumerate the choices of cone angles (or equivalently, singular curvatures) for flat cone metrics on the sphere such that the completion of the relevant moduli space is an orbifold. In his appendix, Thurston stated

These examples were enumerated by a routine computer program, which checks all possibilities having a given least common denominator $q$. The enumeration was not rigorously verified (even though it should not be hard to do so and search more ‘intelligently’ at the same time) but was a simple check of all denominators through 999 in a few minutes of computer time.

How do we know that this enumeration is complete? Is there some condition I'm missing that tells us we won't find orbifold moduli spaces past denominator $q=999$?

How did Thurston do this enumeration so quickly? I've tried to run a similar computer search myself. As best I can figure, the enumeration requires generating all partitions of integers. There are many, many partitions of integers (apparently the number of partitions of $n$ grows like $\exp(\sqrt n)/n$). Thurston was doing this in the 1980's. In my computer search on a modern day laptop, I wasn't able to get past denominators in the 30's even with many hours of home computer time (though I'm certainly no expert programmer).

[1] Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986). ZBL0615.22008.

[2] Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, Publ. Math., Inst. Hautes Étud. Sci. 63, 91-106 (1986). ZBL0615.22009.

[3] Thurston, William P., Shapes of polyhedra and triangulations of the sphere, Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511-549 (1998). ZBL0931.57010.

In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used the perspective of flat structures to get the same results [3] (and apparently corrected the Deligne–Mostow list by slightly extending it, according to another paper of Mostow). In the Thurston perspective, the idea is to enumerate the choices of cone angles (or equivalently, singular curvatures) for flat cone metrics on the sphere such that the completion of the relevant moduli space is an orbifold. In his appendix, Thurston stated

These examples were enumerated by a routine computer program, which checks all possibilities having a given least common denominator $q$. The enumeration was not rigorously verified (even though it should not be hard to do so and search more ‘intelligently’ at the same time) but was a simple check of all denominators through 999 in a few minutes of computer time.

How do we know that this enumeration is complete? Is there some condition I'm missing that tells us we won't find orbifold moduli spaces past denominator $q=999$?

How did Thurston do this enumeration so quickly? I've tried to run a similar computer search myself. As best I can figure, the enumeration requires generating all partitions of integers. There are many, many partitions of integers (apparently the number of partitions of $n$ grows like $\exp(\sqrt n)/n$). Thurston was doing this in the 1980's. In my computer search on a modern day laptop, I wasn't able to get past denominators in the 30's even with many hours of home computer time (though I'm certainly no expert programmer).

Additional thought: One way to cut down the search time is to realize that all of the boundary strata, which correspond to collision of two cone points, must necessarily be smaller moduli spaces that are also orbifolds. (This idea appears for intance as Lemma 2.4 in Mostow's later paper [4].) So e.g. you could try to enumerate all 5-part partitions, then examine only the 6-part partitions which are refinements of those, etc. I imagine this could speed up the search.

[1] Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986). ZBL0615.22008.

[2] Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, Publ. Math., Inst. Hautes Étud. Sci. 63, 91-106 (1986). ZBL0615.22009.

[3] Thurston, William P., Shapes of polyhedra and triangulations of the sphere, Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511-549 (1998). ZBL0931.57010.

[4] Mostow, G. D., On discontinuous action of monodromy groups on the complex n-ball, J. Am. Math. Soc. 1, No. 3, 555-586 (1988). ZBL0657.22014.

In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\text{PU}(1,n)$ (see [1] and [2]). Thurston used the perspective of flat structures to get the same results [3] (and apparently corrected the Deligne-Mostow list by slightly extending it, according to another paper of Mostow). In the Thurston perspective, the idea is to enumerate the choices of cone angles (or equivalently, singular curvatures) for flat cone metrics on the sphere such that the completion of the relevant moduli space is an orbifold. In his appendix, Thurston stated

These examples were enumerated by a routine computer program, which checks all possibilities having a given least common denominator $q$  . The enumeration was not rigorously verified (even though it should not be hard to do so and search more ‘intelligently’ at the same time) but was a simple check of all denominators through 999 in a few minutes of computer time.

How do we know that this enumeration is complete? Is there some condition I'm missing that tells us we won't find orbifold moduli spaces past denominator $q=999$?

How did Thurston do this enumeration so quickly? I've tried to run a similar computer search myself. As best I can figure, the enumeration requires generating all partitions of integers. There are many, many partitions of integers (apparently the number of partitions of $n$ grows like $\exp(\sqrt n)/n$). Thurston was doing this in my 1980's. In my computer search on a modern day laptop, I wasn't able to get past denominator in the 30's even with many hours of home computer time (though I'm certainly no expert programmer).

[1] Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986). ZBL0615.22008.

[2] Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, Publ. Math., Inst. Hautes Étud. Sci. 63, 91-106 (1986). ZBL0615.22009.

[3] Thurston, William P., Shapes of polyhedra and triangulations of the sphere, Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511-549 (1998). ZBL0931.57010.

In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used the perspective of flat structures to get the same results [3] (and apparently corrected the Deligne–Mostow list by slightly extending it, according to another paper of Mostow). In the Thurston perspective, the idea is to enumerate the choices of cone angles (or equivalently, singular curvatures) for flat cone metrics on the sphere such that the completion of the relevant moduli space is an orbifold. In his appendix, Thurston stated

These examples were enumerated by a routine computer program, which checks all possibilities having a given least common denominator $q$. The enumeration was not rigorously verified (even though it should not be hard to do so and search more ‘intelligently’ at the same time) but was a simple check of all denominators through 999 in a few minutes of computer time.

How do we know that this enumeration is complete? Is there some condition I'm missing that tells us we won't find orbifold moduli spaces past denominator $q=999$?

How did Thurston do this enumeration so quickly? I've tried to run a similar computer search myself. As best I can figure, the enumeration requires generating all partitions of integers. There are many, many partitions of integers (apparently the number of partitions of $n$ grows like $\exp(\sqrt n)/n$). Thurston was doing this in the 1980's. In my computer search on a modern day laptop, I wasn't able to get past denominators in the 30's even with many hours of home computer time (though I'm certainly no expert programmer).

[1] Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986). ZBL0615.22008.

[2] Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, Publ. Math., Inst. Hautes Étud. Sci. 63, 91-106 (1986). ZBL0615.22009.

[3] Thurston, William P., Shapes of polyhedra and triangulations of the sphere, Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511-549 (1998). ZBL0931.57010.