Abstract
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov–Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension \(n\) defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
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Acknowledgments
This research was carried out as part of the author’s dissertation project at the University of Notre Dame, with the ever-helpful advice of Karsten Grove. During that time, the author was supported in part by a grant from the U.S. National Science Foundation. The author is grateful to Vitali Kapovitch and Curtis Pro for interesting and helpful conversations on this subject, and to Karsten Grove for pointing out the possibility of using [15] to prove Proposition 3.2.
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Harvey, J. Equivariant Alexandrov Geometry and Orbifold Finiteness. J Geom Anal 26, 1925–1945 (2016). https://doi.org/10.1007/s12220-015-9614-6
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DOI: https://doi.org/10.1007/s12220-015-9614-6