Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

Abstract

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n − 1, 1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n F n−1 are thin.

On orthogonal hypergeometric groups of degree five

“…It is clear that there are finitely many pairs of degree five integer coefficient polynomials having roots of unity as their roots; among these pairs we find that there are 77 pairs (cf. Tables 1,2,3,4,5,6,7) which satisfy the conditions of the last paragraph. Now, we consider the question to determine the arithmeticity or thinness of the associated orthogonal hypergeometric groups Γ(f, g).…”

“…For the orthogonal cases: when the quadratic form Q has signature (n − 1, 1), Fuchs, Meiri and Sarnak give 7 (infinite) families (depending on n odd and ≥ 5) of thin Γ(f, g) in [5]; when the quadratic form Q has signature (p, q) with p, q ≥ 2, 11 (infinite) families of arithmetic Γ(f, g) are given by Venkataramana in [16]; an example of thin Γ(f, g) in O(2, 2) is given by Fuchs in [4]; and 2 examples of arithmetic Γ(f, g) in O (3,2) are given by Singh in [11], which deals with the 14 orthogonal hypergeometric groups of degree five with a maximally unipotent monodromy, and these cases were inspired by the 14 symplectic hypergeometric groups associated to Calabi-Yau threefolds [10].…”

Commensurability and Arithmetic Equivalence for Orthogonal Hypergeometric Monodromy Groups

“…In There are a further 7 hypergeometric groups, which are of type O(4, 1), and whose thin-ness follows from [FMS14] (cf. [BS15, Table 1]).…”

“…In the following table, by the preceeding argument, any forms lying in different equivalence classes (with respect to their Hasse-Witt invariants) cannot intersect in a Zariski dense subgroup of SO q . α, β 1st row of Q Hasse invariants Nature α = 0, 0, 0, 1 3 , 2 3 β = 1 2 , 1 12 , 5 12 , 7 12 , 11 12 (7, 5, 3, 1, −5) (1, 1, 1, 1, 1) Thin [FMS14] α = 0, 0, 0, 1 3 , 2 3 β = 1 2 , 1 8 , 3 8 , 5 8 , 7 8 (7, 1, −1, 1, −9) Thin [FMS14] α = 0, 0, 0, 1 4 , 3 4 β = 1 2 , 1 10 , 3 10 , 7 10 , 9 10 (19, 13, 3, −3, −13) Thin [FMS14] α = 0, 1 8 , 3 8 , 5 8 , 7 8 β = 1 2 , 1 5 , 2 5 , 3 5 , 4 5 (19, −11, −1, 9, −21) Thin [FMS14] α = 0, 0, 0, 1 6 , 5 6 β = 1 2 , 1 10 , 3 10 , 7 10 , 9 10 (67, 53, 19, −19, −53) (−1, −1, 1, 1, 1) Thin [FMS14] α = 0, 1 3 , 2 3 , 1 6 , 5 6 β = 1 2 , 1 5 , 2 5 , 3 5 , 4 5 (19, −1, −11, −1, −11) Thin [FMS14] α = 0, 1 4 , 3 4 , 1 6 , 5 6 β = 1 2 , 1 5 , 2 5 , 3 5 , 4 5 (67, 17, −53, −43, 7) (1, −1, −1, 1, 1) Thin [FMS14] (6, 0, −2, 0, −10) ?? α = 0, 0, 0, 1 3 , 2 3 β = 1 2 , 1 2 , 1 2 , 1 4 , 3 4 (16, −6, −8, 10, −16) ??…”

WHAT IS...a Thin Group?

“…It was shown in [BT14] that this group is thin. For general monodromy groups, determining who is thin or not is wide open; see related work in [Ven14] and [FMS14], as well as the discussion in [Sar14, §3.5]. generate a group Γ < GL 4 (Z).…”