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Translation surfaces and the curve graph in genus two

Duc-Manh Nguyen

Algebraic & Geometric Topology 17 (2017) 2177–2237

Let S be a (topological) compact closed surface of genus two. We associate to each translation surface (X,ω) Ω2 = (2) (1,1) a subgraph Ĉcyl of the curve graph of S. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic or by a concatenation of two parallel saddle connections (satisfying some additional properties) on X. The subgraph Ĉcyl is by definition GL+(2, )–invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that Ĉ cyl is always connected and has infinite diameter. The group Aff+(X,ω) of affine automorphisms of (X,ω) preserves naturally Ĉ cyl, we show that Aff+(X,ω) is precisely the stabilizer of Ĉ cyl in Mod(S). We also prove that Ĉ cyl is Gromov-hyperbolic if (X,ω) is completely periodic in the sense of Calta.

It turns out that the quotient of Ĉ cyl by Aff+(X,ω) is closely related to McMullen’s prototypes in the case that (X,ω) is a Veech surface in (2). We finally show that this quotient graph has finitely many vertices if and only if (X,ω) is a Veech surface for (X,ω) in both strata (2) and (1,1).

translation surface, curve complex, Gromov hyperbolicity
Mathematical Subject Classification 2010
Primary: 51H20
Secondary: 54H15
Received: 31 March 2016
Revised: 30 September 2016
Accepted: 27 October 2016
Published: 3 August 2017
Duc-Manh Nguyen
Institut de Mathématiques de Bordeaux
Université de Bordeaux, CNRS UMR 5251
351, Cours de la Libération
F-33405 Talence Cedex