Convergence of isometries, with semicontinuity of symmetry of Alexandrov spaces

Harvey, John

doi:10.1090/proc/12994

Convergence of isometries, with semicontinuity of symmetry of Alexandrov spaces
HTML articles powered by AMS MathViewer

by John Harvey PDF

Proc. Amer. Math. Soc. 144 (2016), 3507-3515 Request permission

Abstract:

The equivariant Gromov–Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.

References

Mladen Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), no. 1, 143–161. MR 932860, DOI 10.1215/S0012-7094-88-05607-4
Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
John Ennis and Guofang Wei, Describing the universal cover of a compact limit, Differential Geom. Appl. 24 (2006), no. 5, 554–562. MR 2254056, DOI 10.1016/j.difgeo.2006.05.008
Kenji Fukaya, Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N.S.) 12 (1986), no. 1, 121–160. MR 914311, DOI 10.4099/math1924.12.121
Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333. MR 1185120, DOI 10.2307/2946606
Fernando Galaz-Garcia and Luis Guijarro, Isometry groups of Alexandrov spaces, Bull. Lond. Math. Soc. 45 (2013), no. 3, 567–579. MR 3065026, DOI 10.1112/blms/bds101
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534, DOI 10.1007/BF02698687
Karsten Grove, Hermann Karcher, and Ernst A. Ruh, Group actions and curvature, Invent. Math. 23 (1974), 31–48. MR 385750, DOI 10.1007/BF01405201
Karsten Grove and Peter Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), no. 1, 195–206. MR 951512, DOI 10.2307/1971439
Karsten Grove and Peter Petersen V, Volume comparison à la Aleksandrov, Acta Math. 169 (1992), no. 1-2, 131–151. MR 1179015, DOI 10.1007/BF02392759
Karsten Grove and Frederick Wilhelm, Hard and soft packing radius theorems, Ann. of Math. (2) 142 (1995), no. 2, 213–237. MR 1343322, DOI 10.2307/2118635
John Harvey, Equivariant Alexandrov geometry and orbifold finiteness, J. Geom. Anal. (2015), 1–21, published online.
Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269–316. MR 1865418, DOI 10.1007/s002090100252
Frédéric Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988), no. 1, 53–80 (French). MR 958589, DOI 10.1007/BF01394344
G Perelman, Alexandrov’s spaces with curvatures bounded from below II, Preprint available at http://www.math.psu.edu/petrunin/papers/papers.html, 1991.
Curtis Pro, Michael Sill, and Frederick Wilhelm, Crosscap stability, Adv. Geom. (2015), to appear.
—, The diffeomorphism type of manifolds with almost maximal volume, Comm. Anal. Geom. (2015), to appear.