Quaternionic covers and monodromy of the Kontsevich–Zorich cocycle in orthogonal groups

Abstract

We give an example of a Teichm\uller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index, this is the same as $SU(3,1)$ in its second exterior power representation. The example is constructed using origamis (i.e. square-tiled surfaces). It can be generalized to give monodromy inside the group $SO^*(2n)$ for all $n$, but in the general case the monodromy might split further inside the group. Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of Matheus-Yoccoz-Zmiaikou.