Geodesic
Diagonal complexes for surfaces of finite type and surfaces with involution
Algebra i analiz, Tome 33 (2021) no. 3, pp. 51-72.

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Two constructions are studied that are inspired by the ideas of a recent paper by the authors. — The diagonal complex $\mathcal{D}$ and its barycentric subdivision $\mathcal{BD}$ related to an oriented surface of finite type $F$ equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes. — The symmetric diagonal complex $\mathcal{D}^{\text{inv}}$ and its barycentric subdivision $\mathcal{BD}^{\text{inv}}$ related to a symmetric (=with an involution) oriented surface $F$ equipped with a number of (symmetrically placed) labeled marked points. The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.
Keywords: moduli space, ribbon graphs, curve complex, associahedron. 
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G. Panina; J. Gordon. Diagonal complexes for surfaces of finite type and surfaces with involution. Algebra i analiz, Tome 33 (2021) no. 3, pp. 51-72. http://geodesic.mathdoc.fr/item/AA_2021_33_3_a2/

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