Skip to main content
×
Home
    • Aa
    • Aa

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

  • Marcin Sabok (a1) (a2)
Abstract

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$ , due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. G. Atim and R. R. Kallman , The infinite unitary and related groups are algebraically determined Polish groups, Topology Appl. 159(12) (2012), 28312840.

I. Ben Yaacov , A. Berenstein , C. W. Henson and A. Usvyatsov , Model theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Vol. 2, London Mathematical Society Lecture Note Series, Volume 350, pp. 315427 (Cambridge University Press, Cambridge, 2008).

I. Ben Yaacov and T. Tsankov , Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups, Trans. Amer. Math. Soc. 368(11) (2016), 82678294.

R. M. Dudley , Continuity of homomorphisms, Duke Math. J. 28 (1961), 587594.

E. Glasner , The group Aut(𝜇) is Roelcke precompact, Canad. Math. Bull. 55(2) (2012), 297302.

R. R. Kallman , Uniqueness results for groups of measure preserving transformations, Proc. Amer. Math. Soc. 95(1) (1985), 8790.

R. R. Kallman , Uniqueness results for homeomorphism groups, Trans. Amer. Math. Soc. 295(1) (1986), 389396.

A. S. Kechris , Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, Volume 160 (American Mathematical Society, Providence, RI, 2010).

A. S. Kechris and B. D. Miller , Topics in Orbit Equivalence, Lecture Notes in Mathematics, Volume 1852 (Springer, Berlin, 2004).

J. Kittrell and T. Tsankov , Topological properties of full groups, Ergod. Th. & Dynam. Sys. 30(2) (2010), 525545.

M. Malicki , Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures, J. Symbolic Logic 81(3) (2016), 876886.

M. Malicki , The automorphism group of the Lebesgue measure has no non-trivial subgroups of index <2𝜔 , Colloq. Math. 133(2) (2013), 169174.

K. Mann , Automatic continuity for homeomorphism groups and applications. With an appendix by Frédéric Le Roux and Mann, Geom. Topol. 20(5) (2016), 30333056.

V. G. Pestov , A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space, Topology Appl. 155(14) (2008), 15611575.

L. Ribes and P. A. Zalesskii , On the profinite topology on a free group, Bull. Lond. Math. Soc. 25(1) (1993), 3743.

C. Rosendal , Automatic continuity in homeomorphism groups of compact 2-manifolds, Israel J. Math. 166 (2008), 349367.

C. Rosendal , Automatic continuity of group homomorphisms, Bull. Symbolic Logic 15(2) (2009), 184214.

C. Rosendal , Finitely approximable groups and actions. Part I: the Ribes–Zalesskiĭ property, J. Symbolic Logic 76(4) (2011), 12971306.

K. Slutsky , Automatic continuity for homomorphisms into free products, J. Symbolic Logic 78(4) (2013), 12881306.

S. Solecki , Extending partial isometries, Israel J. Math. 150 (2005), 315331.

L. Stojanov , Total minimality of the unitary groups, Math. Z. 187(2) (1984), 273283.

K. Tent and M. Ziegler , The isometry group of the bounded Urysohn space is simple, Bull. Lond. Math. Soc. 45(5) (2013), 10261030.

K. Tent and M. Ziegler , On the isometry group of the Urysohn space, J. Lond. Math. Soc. (2) 87(1) (2013), 289303.

T. Tsankov , Automatic continuity for the unitary group, Proc. Amer. Math. Soc. 141(10) (2013), 36733680.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 56 *
Loading metrics...

* Views captured on Cambridge Core between 11th April 2017 - 24th June 2017. This data will be updated every 24 hours.