Abstract
We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by R C D ∗-spaces and, under extra assumptions, by C D-spaces, C D ∗ P-spaces. However, we show that the M C C P-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.
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Acknowledgements
The author would like to express his gratitude to Jürgen Jost for his crucial support and valuable advice. He is also very grateful to Rostislav Matveev, and Jim Portegies for their essential and friendly involvement in this work.
He also thanks Fernando Galaz-Garcia, Yu Kitabeppu, Martin Kell, and Tapio Rajala for, among other things, discussions that led to Remark 4.6, and Fabio Cavalletti and Andrea Mondino for bringing [5] to his attention. Lastly, the author is thankful to Nidhi Kaihnsa, Niccolò Pederzani, and Ruijun Wu for commenting on earlier versions of the manuscript, and to the anonymous reviewers for their comments. This work was supported by the IMPRS program of the Max Planck Institute for Mathematics in the Sciences and partially by CONACYT. Open access funding provided by Max Planck Society.
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Sosa, G. The Isometry Group of an RCD ∗ Space is Lie. Potential Anal 49, 267–286 (2018). https://doi.org/10.1007/s11118-017-9656-4
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DOI: https://doi.org/10.1007/s11118-017-9656-4