Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Isometry groups of separable metric spaces

MACIEJ MALICKIa1 p1 and SŁAWOMIR SOLECKIa1

a1 Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA. e-mail: mamalicki@gmail.com; ssolecki@math.uiuc.edu

Abstract

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

(Received November 27 2006)

(Revised February 26 2008)

Correspondence:

p1 Current address: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland.

Footnotes

† Supported by NSF grant DMS-0700841.