# Relations between some works by Deligne-Mostow and Thurston

Happy new year 2016!

A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. IHES 1986) and the work of Thurston [T] on moduli spaces of flat stuctures with prescibed conical singularities on the Riemann sphere (Geom. Topol. Monogr. 1998). By different approachs, Deligne and Mostow on the one hand and Thurston on the other obtain some similar results, in particular a list of 94 complex hyperbolic orbifolds, some of which are non-arithmetic.

Thurston's paper [T] has been published more than 10 years after [DM] has been written (the latter has been received by the IHES editorial board in 1983). Hence it seems at first sight that Thurston (who refers to Picard and to [DM] and [M]) just gives a geometrical reinterpretation (in terms of flat structures with conical singularities on the Riemann Sphere) of some of the results previously obtained by Deligne and Mostow.

However [T] is a revised version of a preprint titled Shapes of polyhedra' dating 1987. This leads me and my coworker to wonder what were the respective motivations of Deligne and Mostow and of Thurston in the papers [DM] and [T] ?

It seems that Deligne’s motivations were not at all influenced by Thurston’s (maybe it was a bit different for Mostow but we know nothing about this). Indeed, [DM] is presented as a continuation of some classical works by Schwarz, Picard and Levavasseur and more recent ones by Terada about the monodromy groups of hypergeometric functions and there is no reference to any work by Thurston whatsoever. Moreover, Deligne wasn't aware of Thurston's paper [T] when my coworker contacted him (recently) in order to learn more about the genesis of [DM].

So, one can ask more precise forms of the above question:

(1) From where came the motivations of Thurston when he was working on the material of [T] ?

(2) When did Thurston begin to work on moduli spaces of singular flat structures on the Riemann sphere?

(3) Was he aware of or influenced by the work by Deligne and Mostow at that time?

Thurston's 1987 preprint could possibly put some light on these questions but we were unable to find a version of it. We would be grateful is someone could provide us an electronic copy of this document.

Thanks in advance for any help.

References:

[DM] Deligne P., Mostow G., Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89.

[M] Mostow G., Generalized Picard lattices arising from half-integral conditions.
Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106.

[T] Thurston W., Shapes of polyhedra and triangulations of the sphere.
InThe Epstein birthday schrift', 511–549, Geom. Topol. Monogr. 1, 1998.

• Short a seance with Bill Thurston's spirit, how could you possibly answer questions (1) and (2)? The answer to (3) is YES. Jan 5 '16 at 13:05
• I don't know Thurston's precise motivation, but I was at Davis when he was revising this paper (I don't have a copy of the original, which had fewer examples than the final version). Part of the motivation of Thurston came from finding parameterizations of the set of triangulations of a sphere where all vertices have degree ≤6≤6. In turn, I think this was motivated by his investigation of triangulations of 3-manifolds in which all the vertex links were sphere triangulations of this sort (see sciencedirect.com/science/article/pii/S0040938302001003). Jan 5 '16 at 13:13