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

TL;DR

This paper constructs an infinite family of discrete subgroups of Sp(4,R) with special properties, using hypergeometric monodromy and ping-pong dynamics, revealing new geometric structures and domains of discontinuity.

Contribution

It introduces a novel family of hypergeometric monodromy groups in Sp(4,R) with explicit geometric and dynamical properties, expanding understanding of hypergeometric differential equations and their monodromy.

Findings

Constructed infinite family of discrete subgroups in Sp(4,R)

Established ping-pong dynamics on cones for these groups

Connected cones with crooked surfaces to identify domains of discontinuity

Abstract

We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ 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 $N\geq 4$. 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.