Boundary of the pyramidal equisymmetric locus of $${\mathcal M}_n$$ M n
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Raquel Díaz
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
· Víctor González-Aguilera
Boundary of the pyramidal equisymmetric locus of Mn Received: 13 May 2019 / Accepted: 17 December 2019 n is a compactification of moduli space Mn Abstract. The augmented moduli space M obtained by adding stable hyperbolic surfaces. The different topological types of the added n . Let Pn ⊂ Mn be the pyramidal locus in stable surfaces produce a stratification of ∂ M moduli space, i.e., the set of hyperbolic surfaces of genus n such that the topological action of its preserving-orientation isometry group is the pyramidal action of the dihedral group Dn . The purpose of this paper is to state the complete list of strata in the boundary of Pn .
1. Introduction Let Mn be the moduli space of genus n ≥ 2, i.e., the space of hyperbolic surfaces of n , introduced by Abikoff genus n up to isometry. The augmented moduli space M [1], is a compactification of Mn obtained by adding stable hyperbolic surfaces. n . Since The topological type of the stable surfaces added give a stratification of M topological stable surfaces are codified by their dual stable graphs (see Sect. 2.1 n and for the definitions), this gives a bijection between the set of strata of M the isomorphism classes of stable graphs of genus n. In this stratification, Mn is also a stratum, the one corresponding to the stable graph consisting on a unique vertex with weight n and no edges. In the sequel, we will refer to these strata as n . As notation, given a stable graph G of genus n, the the topological strata of M stratum corresponding to this graph is denoted by S(G). On the other hand, in [2], Broughton analyzed another stratification of moduli space, given by the topological action of the preserving-orientation isometry group of the points of Mn . That is, two hyperbolic surfaces X, X are in the same equisymmetric stratum (or locus) if the action of Iso+ (X ) on X is topologically equivalent to the action of Iso+ (X ) on X . In order to give the topological class of an action of a group G on a surface Sn , it suffices to give an epimorphism : π1 (O) → G, The first author was partially supported by the Project MTM2012-31973 and by Project MTM2017-89420-P. The second author was partially supported by Project PIA, ACT 1415. R. Díaz (B): Departamento de Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid, Spain. e-mail: [email protected] V. González-Aguilera: Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile. e-mail: [email protected] Mathematics Subject Classification: Primary 32G15; Secondary 14H10
https://doi.org/10.1007/s00229-019-01175-0
R. Díaz, V. González-Aguilera
where O is an orbifold and the kernel of is isomorphic to π1 (Sn ). Thus, an epimorphism as above determines an equisymmetric locus of Mn , which will be denoted Ln (, O, G). Let us fix an equisymmetric locus L = Ln (, O, G) of Mn . By ∂L we mean n \Mn n . If S(G) is a topological
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