## Algebraic & Geometric Topology

### Translation surfaces and the curve graph in genus two

Duc-Manh Nguyen

#### Abstract

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,ω) ∈ Ωℳ2 = ℋ(2) ⊔ℋ(1,1)$ 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 $GL+(2, ℝ)$–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 $ℋ(2)$. 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 $ℋ(1,1)$.

#### Article information

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

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

https://projecteuclid.org/euclid.agt/1510841441

Digital Object Identifier
doi:10.2140/agt.2017.17.2177

Mathematical Reviews number (MathSciNet)
MR3685606

Zentralblatt MATH identifier
1373.51006

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841441

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