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Let $G$ be a group that acts properly discontinuously, co-compactly on a locally finite CW-complex $Z$. Does there exist a metric on $Z$ such that $Z$ becomes a geodesic metric space with that metric, and the action of $G$ becomes geometric (proper discontinuous,co-compact action by isometries) ? In this paper https://arxiv.org/pdf/1707.07760.pdf, using proposition 6.2 we have a proper metric on $Z$ such that action of $G$ on $Z$ is geometric but it is not clear to me that $Z$ becomes geodesic metric space.

Thank for any help in advance.

Yogi
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  • Do you also assume that the action is free? If so, the result can be derived from the available literature. If not, my guess is that it still holds, but it would be a research paper. – Moishe Kohan Jul 17 '21 at 04:27
  • @Moishe Kohan; No, I was not assuming that action is free. But if we assume that action is free, how will this follow? Can you please give a reference? – Yogi Jul 17 '21 at 09:16
  • For free actions one uses Moise's theorem: Since $Z/G$ is a locally connected compact metrizable space, it admits a geodesic metric, see here and my addendum here. Now, lift this metric, as a path-metric, to $Z$ and argue again that the lift is geodesic as in my addendum. – Moishe Kohan Jul 17 '21 at 15:01
  • I forgot to add: The same result is independently due to Bing. – Moishe Kohan Jul 17 '21 at 15:11
  • Ok, Thank you for your explanation. – Yogi Jul 18 '21 at 02:27

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