Parallelogram decompositions and generic surfaces in Hhyp(4)

Nguyen, Duc-Manh

doi:10.2140/gt.2011.15.1707

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Parallelogram decompositions and generic surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$

Duc-Manh Nguyen

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 2010

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

Duc-Manh Nguyen
Institut de Mathématiques de Bordeaux, Bat A33 Université Bordeaux 1 351, cours de la Libération F-33405 Talence Cedex France
http://www.math.u-bordeaux1.fr/~nguyendm/

Volume 15, issue 3 (2011)