Isometry groups of proper metric spaces

Niemiec, Piotr

doi:10.1090/S0002-9947-2013-05941-7

Isometry groups of proper metric spaces
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by Piotr Niemiec PDF

Trans. Amer. Math. Soc. 366 (2014), 2597-2623 Request permission

Abstract:

Given a locally compact Polish space $X$, a necessary and sufficient condition for a group $G$ of homeomorphisms of $X$ to be the full isometry group of $(X,d)$ for some proper metric $d$ on $X$ is given. It is shown that every locally compact Polish group $G$ acts freely on $G \times X$ as the full isometry group of $G \times X$ with respect to a certain proper metric on $G \times X$, where $X$ is an arbitrary locally compact Polish space having more than one point such that $(\mathrm {card}(G),\mathrm {card}(X)) \neq (1,2)$. Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space $X$ having more than two points the set of all proper metrics $d$ such that $\mathrm {Iso}(X,d) = \{\mathrm {id}_X\}$ is dense in the space of all proper metrics on $X$.

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