Geometry & Topology 15 (2011) 1707–1747

DOI: 10.2140/gt.2011.15.1707

Abstract

The space H^hyp(4) is the moduli space of pairs (M,ω), where M is a hyperelliptic Riemann surface of genus 3 and ω is a holomorphic 1–form having only one zero. In this paper, we first show that every surface in H^hyp(4) admits a decomposition into parallelograms and simple cylinders following a unique model. We then show that if this decomposition satisfies some irrational condition, then the GL⁺(2, R)–orbit of the surface is dense in H^hyp(4); such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in H^hyp(4) with coordinates in any quadratic field, and there are Thurston–Veech surfaces with trace field of degree three over Q which are generic.

Keywords

translation surface, unipotent flow, dynamics on moduli space

Mathematical Subject Classification

Primary: 51H25

Secondary: 37B05

References

Publication

Received: 6 December 2010
Revised: 12 September 2011
Accepted: 29 August 2011
Published: 1 October 2011
Proposed: Benson Farb
Seconded: Walter Neumann, Joan Birman

Authors