Zorich conjecture for hyperelliptic Rauzy–Veech groups

Abstract

We describe the structure of hyperelliptic Rauzy diagrams and hyperelliptic Rauzy-Veech groups. In particular, this provides a solution of the hyperelliptic cases of a conjecture of Zorich on the Zariski closure of Rauzy-Veech groups.

The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

“…, so that RV (k, )| H k is the explicit finite-index subgroup described in Theorem 2.9 of our previous paper [1].…”

“…Mapping classes attached to the arrows of D d . In this subsection, we essentially review the content of Section 4 of our previous paper [1].…”

“…In this case, the hermitian form has signature (1,1). Denote by U (Q α ) and SU (Q α ) the associated unitary and special unitary groups.…”

Classification of Rauzy–Veech groups: proof of the Zorich conjecture

“…Using the adjacency of strata, we will prove that RV(π)| V contains the Rauzy-Veech group of a connected component of a minimal stratum. We will then analyse which strata are adjacent to each other, which will conclude the proof of Theorem 1.1 since the Rauzy-Veech group of any connected component of a minimal stratum is Zariskidense by the classification of connected components [KZ03], the work on the hyperelliptic case by Avila, Matheus and Yoccoz [AMY17] and Theorem 1.3.…”

“…However, the converse is not true: there are known examples of pinching and twisting groups with small Zariski closure [AMY17, Appendix A]. The pioneering work of Avila, Matheus and Yoccoz [AMY17] shows that this conjecture holds for the particular case of hyperelliptic Rauzy-Veech groups. Their methods also laid the groundwork for further results in this direction.…”