A lantern lemma.

*(English)*Zbl 1022.57001Let \(S\) be an orientable surface and \(\text{Mod}(S)\) the mapping class group of \(S\) (i.e., the group of orientation preserving self-homeomorphisms of \(S\), modulo isotopy). In [Invent. Math. 135, 523-633 (2002; Zbl 0978.57014)], N. V. Ivanov and J. D. McCarty show the existence of an interesting interplay between the algebraic and topological aspects of \(\text{Mod}(S)\), by proving the algebraic characterization of topological reflectiveness, disjointness relation and braid relation between Dehn twists (as far as these topological relations are concerned, see, for example, N. V. Ivanov [Mapping class groups (North-Holland, Amsterdam) (2002; Zbl 1002.57001)]).

The present paper faces – and positively solves – the problem of the existence of a similar characterization of the lantern relation, which is a relation between Dehn twists about curves which lie on a sphere with four punctures (see M. Dehn [Papers on group theory and topology (Springer-Verlag, New York) (1987)] and D. L. Johnson [Proc. Amer. Math. Soc. 75, 119-125 (1979; Zbl 0407.57003)]). Furthermore, the author makes use of results proven by A. Ishida [Proc. Japan Acad. Ser. A Math. Sci. 72, 240-241 (1996; Zbl 0883.57016)] and H. Hamidi-Tehrani [Algebr. Geom. Topol. 2, 1155-1178 (2002; Zbl 1023.57001)] in order to obtain an analogue characterization of another relation involving multitwists, i.e., the 2-chain relation.

Note that the paper cited last independently yields similar results.

The present paper faces – and positively solves – the problem of the existence of a similar characterization of the lantern relation, which is a relation between Dehn twists about curves which lie on a sphere with four punctures (see M. Dehn [Papers on group theory and topology (Springer-Verlag, New York) (1987)] and D. L. Johnson [Proc. Amer. Math. Soc. 75, 119-125 (1979; Zbl 0407.57003)]). Furthermore, the author makes use of results proven by A. Ishida [Proc. Japan Acad. Ser. A Math. Sci. 72, 240-241 (1996; Zbl 0883.57016)] and H. Hamidi-Tehrani [Algebr. Geom. Topol. 2, 1155-1178 (2002; Zbl 1023.57001)] in order to obtain an analogue characterization of another relation involving multitwists, i.e., the 2-chain relation.

Note that the paper cited last independently yields similar results.

Reviewer: Maria Rita Casali (Modena)

##### MSC:

57M07 | Topological methods in group theory |

20F38 | Other groups related to topology or analysis |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57M99 | General low-dimensional topology |

**OpenURL**

##### References:

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[2] | J S Birman, A Lubotzky, J McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107 · Zbl 0551.57004 |

[3] | M Dehn, Papers on group theory and topology, Springer (1987) · Zbl 1264.01046 |

[4] | H Hamidi-Tehrani, Groups generated by positive multi-twists and the fake lantern problem, Algebr. Geom. Topol. 2 (2002) 1155 · Zbl 1023.57001 |

[5] | A Ishida, The structure of subgroup of mapping class groups generated by two Dehn twists, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) 240 · Zbl 0883.57016 |

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[7] | N V Ivanov, J D McCarthy, On injective homomorphisms between Teichmüller modular groups I, Invent. Math. 135 (1999) 425 · Zbl 0978.57014 |

[8] | D L Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979) 119 · Zbl 0407.57003 |

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