Thin monodromy in Sp(4) and Sp(6)

Abstract

We explore the thinness of hypergeometric groups of type $\mathrm{Sp}\left(4\right)$ and $\mathrm{Sp}\left(6\right)$ by applying a new approach of computer-assisted ping-pong. We prove the thinness of 17 hypergeometric groups with maximally unipotent monodromy in $\mathrm{Sp}\left(6\right)$, completing the classification of all 40 such groups into arithmetic and thin cases.

In addition, we establish the thinness of an additional 46 hypergeometric groups in $\mathrm{Sp}\left(6\right)$, and of three hypergeometric groups in $\mathrm{Sp}\left(4\right)$, completing the classification of all $\mathrm{Sp}\left(4\right)$ hypergeometric groups. To the best of our knowledge, this article produces the first 63 examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.