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A note on the curve complex of the 3-holed projective plane. (English) Zbl 1473.57049

This article gives a rather detailed description of the complex of curves of the non-orientable surface \(N_{1,3}\) of genus 1 and with 3 holes.
The author relates (in Proposition 1) the curve complex \(\mathcal{C}(N_{1,3})\) to the Farey complex. Building on this, he gives an explicit exhaustion of \(\mathcal{C}(N_{1,3})\) by finite rigid sets in Theorem 1 (cf. [J. Aramayona and C. J. Leininger, J. Topol. Anal. 5, No. 2, 183–203 (2013; Zbl 1277.57017)], a sub-complex \(X\) of the curve complex of a surface \(S\) is called rigid if every locally injective simplicial map \(X\to\mathcal{C}(S)\) is induced by some homeomorphism of \(S\)), proves that \(\mathcal{C}(N_{1,3})\) is quasi-isometric to a simplicial tree (Theorem 2), and proves that \(\mathcal{C}(N_{1,3})\) is \(3\)-hyperbolic (Proposition 3).

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
20F38 Other groups related to topology or analysis

Citations:

Zbl 1277.57017
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Full Text: arXiv Link

References:

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