# Tessellating the moduli space of strictly convex projective structures on the once-punctured torus

### joint with Stephan Tillmann

Posted as arXiv:1512.04236.

Published in Exp. Math. 28 (3), 369--384, 2019.

This paper generalizes to a higher Teichmüller theory one of the properties of Teichmüller space discovered in the ’80s.

The collection of all ordered $$n$$-tuples of points in the plane is itself a topological space, $$C_n,$$ which we may regard as identical to $$2n$$-dimensional Euclidean space. The symmetric group $$S_n$$ on $$n$$ objects acts naturally on this space by permuting the points. The quotient of this space by this group action is the space of unordered $$n$$-tuples of To some extent it is a matter of taste whether to work with $$C_n$$ considered as a space with an action of $$S_n,$$ or whether instead to work with the quotient as an orbifold. I am most familiar with the $$S_n$$-set approach to these matters.

An analogous choice occurs when considering the several different ways to endow a compact surface with a hyperbolic metric. The space of hyperbolic metrics considered up to isometry ends up being most naturally construed as an orbifold, where the special points correspond to metrics with a nontrivial isometry group. I prefer instead working with Teichmüller space, which is analogous to the space of ordered n-tuples. In brief, a marked hyperbolic metric on a surface $$S$$ is a homeomorphism $$f: X \to S$$ from a surface $$X$$ already endowed with a hyperbolic metric. Two such marked metrics $$f, g$$ are isotopic when, letting $$F$$ be the inverse of $$f,$$ the homeomorphism $$g \circ F$$ is isotopic to the identity. The Teichmüller space of $$S,$$ $$T(S),$$ is then the space of hyperbolic metrics on $$S$$ considered not up to isometry, but up to the stronger equivalence relation of isotopy. It turns out that Teichmüller space too is always homeomorphic to some finite-dimensional Euclidean space (the dimension is $$6g-6$$ where $$g$$ is the genus of the surface). Moreover, there is a natural group action on this space. Let $$Mod(S)$$ be the group of self-homeomorphisms $$f: S \to S,$$ but modded out by the normal subgroup of homeomorphisms isotopic to the identity. This is called the mapping class group of $$S.$$ Then $$Mod(S)$$ turns out to be a countable discrete group that acts naturally, faithfully, and properly discontinuously on $$T(S),$$ just as $$S_n$$ acts naturally on configuration space.

A standard example of a faithful properly discontinuous action is the action of the group $$\mathbb{Z}[i]$$ of Gaussian integers on the complex numbers, by addition. Now, let $$I^2$$ be the unit square of complex numbers with real part and imaginary part at least 0 and less than 1. Then every complex number $$z$$ can be written $$z = s + g,$$ for a unique $$s$$ in $$I^2$$ and $$g$$ a Gaussian integer. Another way to put this is that we can tile all of $$\mathbb{C}$$ by tiles that are images of $$I^2$$ under $$\mathbb{Z}[i].$$ We call $$I^2$$ a fundamental domain for the action of $$\mathbb{Z}[i]$$ on $$\mathbb{C}.$$

Harer and Penner were interested in studying the cohomology of $$Mod(S).$$ One way to get a handle on this is to find a faithful, cellular action of $$Mod(S)$$ on a contractible cell complex. So they were interested in finding a cell decomposition for $$T(S)$$ that was invariant under the action of $$Mod(S).$$ Epstein and the same Penner found such a decomposition for an analogous space of more heavily marked hyperbolic metrics, and Penner used this decomposition to get the desired sort of action of $$Mod(S).$$ I became interested in this subject because an analogous such decomposition is a critical component of the state-of-the-art algorithms in SnapPea for finding hyperbolic structures on three-manifold triangulations.

Recently Cooper and Long generalized this decomposition to Teichmüller-like spaces of a more general class of metrics, called convex projective metrics. They raised the question of whether or not their decomposition was a cell decomposition. Stephan Tillmann and I showed in this paper that their decomposition was indeed a cell decomposition when the surface under consideration was a thrice-punctured sphere or once-punctured torus.