On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates Robert Haraway, Robert L¨owe, Dominic Tate and Stephan Tillmann Abstract Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichm¨ullerspace of punc- tured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's A{coordinates for doubly decorated structures. In addition, we describe a simple, intrinsic edge-flipping algorithm to determine the canonical cell decomposition associated to a point in moduli space, and show that Penner's centres of Teichm¨ullercells are also natural centres of the cells in moduli space. We show that in many cases, the associated holonomy groups are semi-arithmetic. AMS Classification 57M50, 51M10, 51A05, 20H10, 22E40 Keywords convex projective surfaces, Teichm¨ullerspaces, semi-arithmetic group 1 Introduction Classical Teichm¨ullerspace is the moduli space of marked hyperbolic structures of finite volume on a surface. In the case of a punctured surface, many geometrically meaningful ideal cell decompositions for its Teichm¨ullerspace are known. For instance, quadratic differentials are used for the construction attributed to Harer, Mumford and Thurston [14]; hyperbolic geometry and geodesic laminations are used by Bowditch and Epstein [2]; and Penner [22] uses Euclidean cell decompositions associated to the points in decorated Teichm¨ullerspace. The decoration arises from associating a positive real number to each cusp of the surface, and this is used in the key construction of Epstein and Penner [7] that constructs a Euclidean cell decomposition of a decorated and marked hyperbolic surface. All of these decompositions of (decorated) Teichm¨ullerspace are natural in the sense that they are invariant under the action of the mapping class group (and hence descend to a cell decomposition of the moduli space of unmarked structures) and that they do not involve any arbitrary choices. arXiv:1911.04176v1 [math.GT] 11 Nov 2019 A hyperbolic structure is an example of a strictly convex projective structure, and two hyperbolic structures are equivalent as hyperbolic structures if and only if they are equivalent as projective structures. Let Sg;n denote the surface of genus g with n punctures. We will always assume that 2g + n > 2; so that the surface has negative Euler characteristic. Whereas the classical Teichm¨uller 6g−6+2n space Thyp(Sg;n) is homeomorphic with R ; Marquis [20] has shown that the analogous moduli space Tf (Sg;n) of marked strictly convex projective structures of finite volume on Sg;n is homeomorphic 16g−16+6n with R : y There is a natural decorated moduli space Tf (Sg;n), again obtained by associating a positive real number to each cusp of the surface. Cooper and Long [6] generalised the construction of Epstein 1 y and Penner [7], thus associating to each point in the decorated moduli space Tf (S) of strictly convex projective structures of finite volume an ideal cell decomposition of S: Cooper and Long [6] state that y their construction can be used to define a decomposition of the decorated moduli space Tf (S); but that it is not known whether all components of this decomposition are cells. The main result of this paper establishes the fact that this is indeed always the case. This was previously only known in the case where S is the once-punctured torus or a sphere with three punctues [13]. n y As in the classical setting, there is a principal R+ foliated fibration Tf (Sg;n) ! Tf (Sg;n); and different points in a fibre above a point of Tf (Sg;n) may lie in different components of the cell decomposition y of Tf (Sg;n): However, if there is only one cusp, then all points in a fibre lie in the same component, and one obtains a decomposition of Tf (Sg;1): 2 A strictly convex projective surface is a quotient Ω=Γ; where Ω ⊂ RP is a strictly convex domain and Γ is a discrete group preserving Ω: One technical difficulty in working with strictly convex projective structures arises from the fact that as one varies a point in moduli space, not only the associated holonomy group Γ varies, but also the domain Ω varies. The key in our proof is to use a particularly z nice coordinate system for the space Tf (Sg;n) of doubly-decorated strictly convex projective structures n z y due to Fock and Goncharov [8]. This has a principal R+ foliated fibration Tf (Sg;n) ! Tf (Sg;n); so y different points in a fibre above a point of Tf (Sg;n) have the same canonical cell decomposition. z Fock and Goncharov [8] devise parameterisations of Tf (Sg;n) and Tf (Sg;n) by choosing an ideal trian- gulation of the surface and associating to each triangle one positive parameter and to each edge two. We distinguish the edge parameters by associating them to the edge with different orientations. This 16g−16+8n gives a parameter space diffeomorphic with R ; which is given two different interpretations. 2 The X-coordinates arise from flags in the projective plane RP : This is shown in [9] to parameterise the space of strictly convex projective structures on S with a framing at each end. This amounts to a finite branched cover of the space of strictly convex projective structures on S studied by Goldman, and parameterises structures that may have finite or infinite area. The space Tf (Sg;n) is identified 16g−16+8n with a subvariety of R>0 : See [4] for a complete discussion of these facts. 3 The A-coordinates arise from flags in R : Indeed, they describe the lift of the developing map for 2 3 a strictly convex projective structures from RP to R ; and are shown to parameterise the space of finite-area structures with the additional data of a vector and covector decoration at each cusp of S . z We denote this double decorated space by Tf (Sg;n): We give a complete treatment in §3, where we y also show that the space of independent decorated structures Tf (Sg;n)=R>0 can be identified with a z subset of Tf (Sg;n) that is a product of n + 1 open simplices. The details for the relationship between 3 A-coordinates and lifts of developing maps to R are probably well known to experts. In §4, we introduce the key player in our approach, the outitude associated to an edge of the ideal triangulation of S: We show that, using A{coordinates, there is a simple edge flipping algorithm z resulting in the canonical ideal cell decomposition associated to a point in Tf (Sg;n): The outitude is z then used to prove that we obtain a cell decomposition of Tf (Sg;n) in §5. The main idea of the proof z is to determine a natural product structure for each putative cell in the decomposition of Tf (Sg;n), and to show that each level set in this product structure is star-shaped. This is first done for the case of a triangulation (Theorem 5.2), and the proof for more general cell decompositions of S is more involved (Theorem 5.6). In §6, we analyse projective duality in A-coordinates, identify classical Teichm¨ullerspace in these coordinates and show that our cell decomposition is generally not invariant under duality. Benoist 2 and Hulin [1] showed that Tf (Sg;n) is the product of classical Teichm¨ullerspace Thyp(Sg;n) and the vector space of cubic holomorphic differential on the surface with poles of order at most 2 at the cusps. In particular, any cell decomposition of classical Teichm¨ullerspace gives rise to a cell decomposition y z of the spaces Tf (Sg;n), Tf (Sg;n) and Tf (Sg;n). Duality is used to show that our cell decomposition generally does not arise in this way in §7. Penner [22] describes a natural centre for each of the cells in classical Teichm¨ullerspace, and shows that the centres of top-dimensional cells correspond to arithmetic Fuchsian groups. In §8, we show z that Penner's centres are also natural centres of the cells in Tf (Sg;n), and that they correspond to semi-arithmetic Fuchsian groups in many (but not all) cases. For instance, if the associated cell decomposition of the surface only involves polygons with an odd number of sides, then the associated group is semi-arithmetic. The study of higher Teichm¨uller spaces has emerged through the examples of Hitchin components and maximal representations (see [3, 28] for an overview of the field and references). Generalisations from classical to higher Teichm¨ullertheory typically have a geometric interpretation for convex real z z projective structures in the case of Tf (Sg;n) and Tf (Sg;n). The spaces Tf (Sg;n) and Tf (Sg;n) can thus be viewed as stepping stones between classical Teichm¨ullerspace and the higher Teichm¨ullerspaces. It would be interesting to construct a generalisation of the outitude for arbitrary semisimple Lie groups that leads to cell decompositions of all higher Teichm¨ullerspaces. Acknowledgements Research by L¨owe is supported by the DFG via SFB-TRR 109: \Discretization in Geometry and Dynamics". Tate acknowledges support by the Australian Government Research Training Program. Research of Tillmann is supported in part under the Australian Research Council's ARC Future Fellowship FT170100316.