Translation surfaces and the curve graph in genus two
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | NGUYEN, Duc-Manh | |
| dc.date.accessioned | 2024-04-04T03:04:12Z | |
| dc.date.available | 2024-04-04T03:04:12Z | |
| dc.date.issued | 2017 | |
| dc.identifier.issn | 1472-2747 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193132 | |
| dc.description.abstractEn | Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ 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 $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$. | |
| dc.language.iso | en | |
| dc.publisher | Mathematical Sciences Publishers | |
| dc.title.en | Translation surfaces and the curve graph in genus two | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.2140/agt.2017.17.2177 | |
| dc.subject.hal | Mathématiques [math]/Topologie géométrique [math.GT] | |
| dc.subject.hal | Mathématiques [math]/Systèmes dynamiques [math.DS] | |
| dc.identifier.arxiv | 1506.05966 | |
| bordeaux.journal | Algebraic and Geometric Topology | |
| bordeaux.page | 2177 - 2237 | |
| bordeaux.volume | 17 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 4 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01925660 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01925660v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Algebraic%20and%20Geometric%20Topology&rft.date=2017&rft.volume=17&rft.issue=4&rft.spage=2177%20-%202237&rft.epage=2177%20-%202237&rft.eissn=1472-2747&rft.issn=1472-2747&rft.au=NGUYEN,%20Duc-Manh&rft.genre=article |
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