Translation surfaces and the curve graph in genus two

Nguyen, Duc-Manh

doi:10.2140/agt.2017.17.2177

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Translation surfaces and the curve graph in genus two

Duc-Manh Nguyen

Algebraic & Geometric Topology 17 (2017) 2177–2237

DOI: 10.2140/agt.2017.17.2177

Abstract

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X, ω) \in Ω ℳ_{2} = ℋ (2) ⊔ ℋ (1, 1)$ a subgraph ${\hat{C}}_{cyl}$ 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 ${\hat{C}}_{cyl}$ is by definition ${GL}^{+} (2, ℝ)$ –invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that ${\hat{C}}_{cyl}$ is always connected and has infinite diameter. The group ${Aff}^{+} (X, ω)$ of affine automorphisms of $(X, ω)$ preserves naturally ${\hat{C}}_{cyl}$ , we show that ${Aff}^{+} (X, ω)$ is precisely the stabilizer of ${\hat{C}}_{cyl}$ in $Mod (S)$ . We also prove that ${\hat{C}}_{cyl}$ is Gromov-hyperbolic if $(X, ω)$ is completely periodic in the sense of Calta.

It turns out that the quotient of ${\hat{C}}_{cyl}$ by ${Aff}^{+} (X, ω)$ is closely related to McMullen’s prototypes in the case that $(X, ω)$ is a Veech surface in $ℋ (2)$ . 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 $ℋ (2)$ and $ℋ (1, 1)$ .

Keywords

translation surface, curve complex, Gromov hyperbolicity

Mathematical Subject Classification 2010

Primary: 51H20

Secondary: 54H15

References

Publication

Received: 31 March 2016

Revised: 30 September 2016

Accepted: 27 October 2016

Published: 3 August 2017

Authors

Duc-Manh Nguyen
Institut de Mathématiques de Bordeaux Université de Bordeaux, CNRS UMR 5251 351, Cours de la Libération F-33405 Talence Cedex France

Volume 17, issue 4 (2017)