## Algebraic & Geometric Topology

### The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

#### Abstract

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $2π3$ but less than $2π$. 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1323-1343.

Dates
Revised: 26 January 2011
Accepted: 15 February 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715229

Digital Object Identifier
doi:10.2140/agt.2011.11.1323

Mathematical Reviews number (MathSciNet)
MR2801420

Zentralblatt MATH identifier
1233.57010

#### Citation

Cruz-Cota, Aldo-Hilario. The moduli space of hex spheres. Algebr. Geom. Topol. 11 (2011), no. 3, 1323--1343. doi:10.2140/agt.2011.11.1323. https://projecteuclid.org/euclid.agt/1513715229

#### References

• M Boileau, J Porti, Geometrization of 3–orbifolds of cyclic type, Astérisque (2001) 208 Appendix A by Michael Heusener and Porti
• D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence, RI (2001)
• S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
• D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan, Tokyo (2000) With a postface by Sadayoshi Kojima
• A-H Cruz-Cota, Classifying Voronoi graphs of hex sphere, preprint
• W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
• N Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449–473
• F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099
• J Loftin, Flat metrics, cubic differentials and limits of projective holonomies, Geom. Dedicata 128 (2007) 97–106
• M Troyanov, On the moduli space of singular Euclidean surfaces, from: “Handbook of Teichmüller theory Vol I”, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 507–540