A cyclotomic family of thin hypergeometric monodromy groups in $${\text {Sp}}_4({\mathbb {R}})$$

Abstract

We exhibit an infinite family of discrete subgroups of \({{\,\mathrm{\textbf{Sp}}\,}}_4(\mathbb {R})\) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters \(\left( \tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right) \) at infinity and maximal unipotent monodromy at zero, for any integer \(N\ge 4\). Additionally, we relate the cones used for ping-pong in \(\mathbb {R}^4\) with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1).