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

Abstract

ABSTRACT. We give an example of a Teichmüller 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.

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

“…Indeed, as was done by Filip-Forni-Matheus [FFM18], we will show that these groups seem to arise in quaternionic covers of simple square-tiled surfaces. This article is organized as follows.…”

“…The groups in the second item were shown to be realizable by Avila-Matheus-Yoccoz [AMY17]. Moreover, the group SO * (6) in its standard representation (which coincides with SU(3, 1) in its second exterior power representation) is also realizable by the work of Filip-Forni-Matheus [FFM18].…”

Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle

“…After a preliminary version of this paper was circulated, in joint work with Matheus and Forni [FFM15] we found instances with monodromy in the group SO * 6 in its standard representation. This coincides with SU 3,1 in the second exterior power representation.…”

“…Recent joint work with Forni and Matheus [FFM15] exhibits the first examples of monodromy in the group SO * 6 , in the standard representation. This group coincides with SU 3,1 in the second exterior power representation, but the methods developed should in principle lead to monodromy in SO * 2n for any n. Question 5.5 Is it possible that some orthogonal group in the spin representation, or SU p,q in a higher exterior power representation, occur in the algebraic hull of the Kontsevich-Zorich cocycle?…”

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

“…A bit more generally, suppose that a finite group G acts on the real bundle E. This is the situation considered in detail by Matheus-Yoccoz-Zmiaikou [31] and also in [13].…”