Moduli spaces of convex projective structures on surfaces

Abstract

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL 3 (R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.

Espace des modules marqués des surfaces projectives convexes de volume fini

“…Nous montrerons que c'est en fait un homéomorphisme. Foch et Goncharov ont montré dans [10] que l'espaceˇg ;p est homéomorphe à R 16g 16C6p . Nous donnons ici un système de coordonnées "à la Fenchel-Nielsen" surˇf .…”

Polynomial cubic differentials and convex polygons in the projective plane

“…On both sides of the analogy, the Teichmüller space is contractible and smooth, while the quotient moduli space is only an orbifold. (Compare the "toy model" of the space of convex projective structures described in [FG07].) So far we have considered our spaces of polygons to be topological spaces using what could be called the vertex topology, i.e.…”

Central extensions of the Ptolemy-Thompson group and quantized Teichmüller theory

“…As it is well-known (see [31]) there is a group structure underlying the groupoid structure that identifies the Ptolemy group of flips on the Farey triangulation to the Thompson group T of piecewise-PSL(2, Z) homeomorphisms of the circle (see [10]). Our aim is to identify the central extension T of T arising in the dilogarithm representations constructed in ( [19], section 10, [18], section 3). We refer to T as the dilogarithmic central extension of T .…”