Algebraic & Geometric Topology

Translation surfaces and the curve graph in genus two

Duc-Manh Nguyen

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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).

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2177-2237.

Received: 31 March 2016
Revised: 30 September 2016
Accepted: 27 October 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51H20: Topological geometries on manifolds [See also 57-XX]
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]

translation surface curve complex Gromov hyperbolicity


Nguyen, Duc-Manh. Translation surfaces and the curve graph in genus two. Algebr. Geom. Topol. 17 (2017), no. 4, 2177--2237. doi:10.2140/agt.2017.17.2177.

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