Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

TL;DR

This paper provides a criterion to identify when groups generated by Cartan involutions in automorphism groups of rational quadratic forms are thin, using graph theory and Vinberg's hyperbolic reflection groups, with applications to hypergeometric groups.

Contribution

It introduces a new criterion based on graph structures and Vinberg's theory to determine thinness of hyperbolic monodromy groups.

Findings

The criterion effectively identifies many hyperbolic hypergeometric groups as thin.

The approach is robust across various cases of hyperbolic monodromy groups.

Application to groups of signature (n-1,1) demonstrates broad utility.

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.