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

TL;DR

This paper constructs a new example of a Teichmüller curve with monodromy group Zariski closure equal to SO^*(6), revealing novel group structures in the monodromy of origamis and computing related cohomological multiplicities.

Contribution

It provides the first known example of a Teichmüller curve with monodromy group SO^*(6) and generalizes this to higher dimensions, also computing cohomological multiplicities.

Findings

Monodromy group Zariski closure is SO^*(6) in a Teichmüller curve.

Construction of examples using origamis (square-tiled surfaces).

Computed multiplicities of representations in cohomology, answering prior questions.

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.