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

Abstract

For all $d$ belonging to a density-$1/8$ subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group $\mathrm{SO}^*(2d)$ 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 $\mathrm{SO}^*(2d)$ is realizable for every $11 \leq d \leq 299$ such that $d = 3 \bmod 8$, except possibly for $d = 35$ and $d = 203$.