Nielsen realization problem

The Nielsen realization problem is a question asked by Jakob Nielsen (1932, p. 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 group Diff(S) the diffeomorphisms of the surface to itself in isotopy classes to get the mapping class group π₀(Diff(S)). The conjecture asks whether a finite group 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, p. 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 in 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", American Mathematical Society. Bulletin. New Series 2 (3): 452–454, doi:10.1090/S0273-0979-1980-14764-3, ISSN 0002-9904, MR561531 
• Kerckhoff, Steven P. (1983), "The Nielsen realization problem", Annals of Mathematics. Second Series 117 (2): 235–265, doi:10.2307/2007076, ISSN 0003-486X, MR690845 
• Kravetz, Saul (1959), "On the geometry of Teichmüller spaces and the structure of their modular groups", Ann. Acad. Sci. Fenn. Ser. A I No. 278: 35, MR0148906 
• Linch, Michele Regina (1971), ON METRICS IN TEICHMUELLER SPACE, ProQuest LLC, Ann Arbor, MI, MR2620985, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7201352 
• Masur, Howard (1975), "On a class of geodesics in Teichmüller space", Annals of Mathematics. Second Series 102 (2): 205–221, ISSN 0003-486X, JSTOR 1971031, MR0385173 
• Nielsen, Jakob (1932), "Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. III." (in German), Acta Math., Uppsala 58 (1): 87–167, doi:10.1007/BF02547775, Zbl 0004.27501