Nielsen realization problem
Nielsen realization problem The Nielsen realization problem is a question asked by Jakob Nielsen (1932, pp. 147–148) about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by Steven Kerckhoff (1980, 1983).
Statement Given an oriented surface, we can divide the group Diff(S), the group of diffeomorphisms of the surface to itself, into isotopy classes to get the mapping class group π0(Diff(S)). The conjecture asks whether a finite subgroup of the mapping class group of a surface can be realized as the isometry group of a hyperbolic metric on the surface.
The mapping class group acts on Teichmüller space. An equivalent way of stating the question asks whether every finite subgroup of the mapping class group fixes some point of Teichmüller space.
History Jakob Nielsen (1932, pp. 147–148) asked whether finite subgroups of mapping class groups can act on surfaces. Kravetz (1959) claimed to solve the Nielsen realization problem but his proof depended on trying to show that Teichmüller space (with the Teichmüller metric) is negatively curved. Linch (1971) pointed out a gap in the argument, and Masur (1975) showed that Teichmüller space is not negatively curved. Steven Kerckhoff (1980, 1983) gave a correct proof that finite subgroups of mapping class groups can act on surfaces using left earthquakes.
References Kerckhoff, Steven P. (1980), "The Nielsen realization problem", Bollettino dell'American Mathematical Society, Nuova serie, 2 (3): 452–454, doi:10.1090/S0273-0979-1980-14764-3, ISSN 0002-9904, SIG 0561531 Kerckhoff, Steven P. (1983), "The Nielsen realization problem", Annali di matematica, Seconda serie, 117 (2): 235–265, CiteSeerX 10.1.1.353.3593, doi:10.2307/2007076, ISSN 0003-486X, JSTOR 2007076, SIG 0690845 Kravetz, Saul (1959), "On the geometry of Teichmüller spaces and the structure of their modular groups", Anna. Accad. Sci. Fenn. Ser. A I No., 278: 35, SIG 0148906 Linch, Michele Regina (1971), ON METRICS IN TEICHMUELLER SPACE, SIG 2620985 – via ProQuest Masur, Howard (1975), "On a class of geodesics in Teichmüller space", Annali di matematica, Seconda serie, 102 (2): 205–221, doi:10.2307/1971031, ISSN 0003-486X, JSTOR 1971031, SIG 0385173 Nielsen, Jakob (1932), "Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. III.", Acta Math. (in tedesco), 58 (1): 87–167, doi:10.1007/BF02547775, Zbl 0004.27501 Categorie: Geometric topologyHomeomorphismsTheorems in topology
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