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

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.