Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Isometry groups of proper metric spaces

Author: Piotr Niemiec
Journal: Trans. Amer. Math. Soc. 366 (2014), 2597-2623
MSC (2010): Primary 37B05, 54H15; Secondary 54D45
Published electronically: October 28, 2013
MathSciNet review: 3165648
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] H. Abels, A. Manoussos, and G. Noskov, Proper actions and proper invariant metrics, J. Lond. Math. Soc. (2) 83 (2011), no. 3, 619-636. MR 2802502 (2012e:54050),
  • [2] Ahmed Bouziad, Every Čech-analytic Baire semitopological group is a topological group, Proc. Amer. Math. Soc. 124 (1996), no. 3, 953-959. MR 1328341 (96f:22002),
  • [3] Jean Braconnier, Sur les groupes topologiques localement compacts, J. Math. Pures Appl. (9) 27 (1948), 1-85 (French). MR 0025473 (10,11c)
  • [4] Nikolaus Brand, Another note on the continuity of the inverse, Arch. Math. (Basel) 39 (1982), no. 3, 241-245. MR 682451 (84b:22001),
  • [5] D. van Dantzig and B. L. van der Waerden, Über metrisch homogene Räume, Abh. Math. Sem. Hamburg 6 (1928), 367-376.
  • [6] Robert Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372-373. MR 0083681 (18,745d)
  • [7] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321 (91c:54001)
  • [8] Su Gao and Alexander S. Kechris, On the classification of Polish metric spaces up to isometry, Mem. Amer. Math. Soc. 161 (2003), no. 766, viii+78. MR 1950332 (2004b:03067)
  • [9] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin, 1979. Structure of topological groups, integration theory, group representations. MR 551496 (81k:43001)
  • [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II (A Series of Comprehensive Studies in Mathematics, Vol. 152), Springer-Verlag, Berlin, 1997.
  • [11] J.-L. Koszul, Lectures on groups of transformations, Notes by R. R. Simha and R. Sridharan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 32, Tata Institute of Fundamental Research, Bombay, 1965. MR 0218485 (36 #1571)
  • [12] A. Lindenbaum, Contributions à l'étude de l'espace métrique I, Fund. Math. 8 (1926), 209-222.
  • [13] Maciej Malicki and Sławomir Solecki, Isometry groups of separable metric spaces, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 67-81. MR 2461868 (2009j:54057),
  • [14] Antonios Manoussos and Polychronis Strantzalos, On the group of isometries on a locally compact metric space, J. Lie Theory 13 (2003), no. 1, 7-12. MR 1958572 (2004a:54045)
  • [15] Julien Melleray, Compact metrizable groups are isometry groups of compact metric spaces, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1451-1455. MR 2367119 (2008m:54048),
  • [16] Helmut Pfister, Continuity of the inverse, Proc. Amer. Math. Soc. 95 (1985), no. 2, 312-314. MR 801345 (87a:22004),
  • [17] L. Pontrjagin, Topological Groups, Princeton University Press, Princeton, 1946.
  • [18] Raimond A. Struble, Metrics in locally compact groups, Compositio Math. 28 (1974), 217-222. MR 0348037 (50 #535)
  • [19] Robert Williamson and Ludvik Janos, Constructing metrics with the Heine-Borel property, Proc. Amer. Math. Soc. 100 (1987), no. 3, 567-573. MR 891165 (88m:54039),
  • [20] W. Żelazko, A theorem on $ B_{0}$ division algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 373-375 (English, with Russian summary). MR 0125901 (23 #A3198)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37B05, 54H15, 54D45

Retrieve articles in all journals with MSC (2010): 37B05, 54H15, 54D45

Additional Information

Piotr Niemiec
Affiliation: Instytut Matematyki, Wydział Matematyki i Informatyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Keywords: Locally compact Polish group, isometry group, transitive group action, proper metric space, Heine-Borel metric space, proper metric, proper action.
Received by editor(s): January 26, 2012
Received by editor(s) in revised form: July 10, 2012
Published electronically: October 28, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society