Symmetry, isotopy, and irregular covers. (English) Zbl 1419.57006

Summary: We say that a covering space of surfaces \(S\to X\) has the Birman-Hilden property if the subgroup of the mapping class group of \(X\) consisting of mapping classes that have representatives that lift to \(S\) embeds in the mapping class group of \(S\) modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a covering space has this property. We give new explicit examples of irregular branched covering spaces that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.


57M10 Covering spaces and low-dimensional topology
57M60 Group actions on manifolds and cell complexes in low dimensions
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[1] Aramayona, J., Souto, J.: Homomorphisms between mapping class groups. arXiv:1011.1855 (2010) · Zbl 1262.57003
[2] Aramayona, J., Leininger, C.J., Souto, J.: Injections of mapping class groups. Geom. Topol. 13(5), 2523-2541 (2009) · Zbl 1225.57001 · doi:10.2140/gt.2009.13.2523
[3] Berstein, I., Edmonds, A.L.: On the construction of branched coverings of low-dimensional manifolds. Trans. Amer. Math. Soc. 247, 87-124 (1979) · Zbl 0359.55001 · doi:10.1090/S0002-9947-1979-0517687-9
[4] Birman, J.S., Hilden, H.M.: Lifting and projecting homeomorphisms. Arch. Math. (Basel) 23, 428-434 (1972) · Zbl 0247.55001 · doi:10.1007/BF01304911
[5] Birman, J.S., Hilden, H.M.: On isotopies of homeomorphisms of Riemann surfaces. Ann. Math. 2(97), 424-439 (1973) · Zbl 0237.57001 · doi:10.2307/1970830
[6] Birman, J.S., Wajnryb, \[B.: 33\]-fold branched coverings and the mapping class group of a surface. In: Geometry and Topology (College Park, Md., 1983/84). Lecture Notes in Mathematics, vol. 1167, pp. 24-46. Springer, Berlin (1985) · Zbl 0589.57009
[7] Casson, A.J., Bleiler, S.A.: Automorphisms of Surfaces after Nielsen and Thurston. Cambridge University Press, Cambridge (1988) · Zbl 0649.57008 · doi:10.1017/CBO9780511623912
[8] Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton University Press, Princeton (2011) · Zbl 1245.57002 · doi:10.1515/9781400839049
[9] Fuller, T.: On fiber-preserving isotopies of surface homeomorphisms. Proc. Amer. Math. Soc. 129(4), 1247-1254 (2001) · Zbl 0977.57002 · doi:10.1090/S0002-9939-00-05642-2
[10] Harvey, W.J., Korkmaz, M.: Homomorphisms from mapping class groups. Bull. Lond. Math. Soc. 37(2), 275-284 (2005) · Zbl 1066.57020 · doi:10.1112/S0024609304003911
[11] Ivanov, N.V., McCarthy, J.D.: On injective homomorphisms between Teichmüller modular groups. I. Invent. Math. 135(2), 425-486 (1999) · Zbl 0978.57014 · doi:10.1007/s002220050292
[12] Luo, F.: Automorphisms of the complex of curves. Topology 39(2), 283-298 (2000) · Zbl 0951.32012 · doi:10.1016/S0040-9383(99)00008-7
[13] Maclachlan, C., Harvey, W.J.: On mapping-class groups and Teichmüller spaces. Proc. Lond. Math. Soc. (3) 30(4), 496-512 (1975) · Zbl 0303.32020 · doi:10.1112/plms/s3-30.4.496
[14] Paris, L., Berrick, J.A., Gebhardt, V.: Finite index subgroups of mapping class groups. arXiv:1105.2468, May (2011) · Zbl 1294.57014
[15] Rafi, K., Schleimer, S.: Covers and the curve complex. Geom. Topol. 13(4), 2141-2162 (2009) · Zbl 1166.57013 · doi:10.2140/gt.2009.13.2141
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