A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle

Abstract

For all \begin{document}$ d $\end{document} belonging to a density- \begin{document}$ 1/8 $\end{document} subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group \begin{document}$ \mathrm{SO}^*(2d) $\end{document} in its standard representation as the Zariski-closure of a factor of its monodromy. We prove that this conjecture holds for the first elements of this subset, showing that the group \begin{document}$ \mathrm{SO}^*(2d) $\end{document} is realizable for every \begin{document}$ 11 \leq d \leq 299 $\end{document} such that \begin{document}$ d = 3 \bmod 8 $\end{document} , except possibly for \begin{document}$ d = 35 $\end{document} and \begin{document}$ d = 203 $\end{document} .