The moduli space of hex spheres

Cruz-Cota, Aldo-Hilario

doi:10.2140/agt.2011.11.1323

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The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

Algebraic & Geometric Topology 11 (2011) 1323–1343

DOI: 10.2140/agt.2011.11.1323

Abstract

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2 π}{3}$ but less than $2 π$ . We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties:

(1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$ ;

(2) it embeds isometrically in the 3–dimensional Euclidean space as the boundary of a tetrahedron; and

(3) there is a simple closed geodesic $γ$ in $M$ such that a fractional Dehn twist along $γ$ converts $M$ to the double of a parallelogram.

Keywords

singular Euclidean surfaces, moduli spaces

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 57M15

References

Publication

Received: 31 October 2010

Revised: 26 January 2011

Accepted: 15 February 2011

Published: 11 May 2011

Authors

Aldo-Hilario Cruz-Cota
Department of Mathematics Grand Valley State University Allendale MI 49401-9401 USA

Volume 11, issue 3 (2011)