Tessellating the Moduli Space of Strictly Convex Projective Structures on the Once-Punctured Torus

ABSTRACT

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of marked strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of marked strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.

Keywords:

• projective surface
• cell decomposition
• moduli space
• convex hull

AMS Subject Classification:

• 57M50
• 57N05
• 14H15

Acknowledgments

The authors thank the two anonymous referees for suggestions and corrections that helped improve this paper.

Notes

1 We wish to emphasize that we make the natural choice of using a two-letter acronym here.

2 At the time the final version of this paper was prepared, qepcad was no longer supported by Sage.

3 Upon submission of our paper, Sage would run the above code with the given output. However, upon revision, it would not do so, as qepcad is not supported in the most recent version of Sage.

Additional information

Funding

This research was partially supported by Australian Research Council under grant DP140100158.