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A lantern lemma. (English) Zbl 1022.57001
Let $$S$$ 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.

##### 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
##### Keywords:
mapping class group; Dehn twist; lantern relation
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##### References:
 [1] , Travaux de Thurston sur les surfaces, Société Mathématique de France (1991) 1 · Zbl 0731.57001 [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 [6] N V Ivanov, Mapping class groups, North-Holland (2002) 523 · Zbl 1002.57001 [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 [9] J D McCarthy, Automorphisms of surface mapping class groups. A recent theorem of N Ivanov, Invent. Math. 84 (1986) 49 · Zbl 0594.57007 [10] Y N Minsky, A geometric approach to the complex of curves on a surfaceuller spaces (Katinkulta, 1995)”, World Sci. Publ., River Edge, NJ (1996) 149 · Zbl 0937.30027
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