Finitely approximable groups and actions. I: The Ribes-Zalesskiĭ property. (English) Zbl 1250.03085

Rosendal, Christian

J. Symb. Log. 76, No. 4, 1297-1306 (2011).

Summary: We investigate extensions of S. Solecki’s theorem on closing off finite partial isometries of metric spaces [S. Solecki, Isr. J. Math. 150, 315–331 (2005; Zbl 1124.54012)] and obtain the following exact equivalence: any action of a discrete group \(\Gamma\) by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of \(\Gamma\) is closed in the profinite topology on \(\Gamma\).

MSC:

03E15	Descriptive set theory
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54H20	Topological dynamics (MSC2010)

Keywords:

Urysohn metric space; profinite topology; Ribes-Zalesskiĭ theorem

Citations:

Zbl 1124.54012

References:

[1]	DOI: 10.1142/S0218196701000449 · Zbl 1024.20022
[2]	Forum Mathematician 17 pp 513– (2005)
[3]	DOI: 10.1142/S0218196791000079 · Zbl 0722.20039
[4]	Commentationes Mathematicae Universitatis Carotinae 31 pp 181– (1990)
[5]	DOI: 10.1007/BF02762385 · Zbl 1124.54012
[6]	DOI: 10.1090/S0002-9947-1949-0032642-4
[7]	Finitely approximable groups and actions. Part II: Generic representations 76 pp 1307– (2011) · Zbl 1250.03086
[8]	DOI: 10.1112/blms/23.4.356 · Zbl 0754.20007
[9]	DOI: 10.1007/BF01305233 · Zbl 0767.05053
[10]	DOI: 10.1090/S0002-9947-99-02374-0 · Zbl 0947.20018
[11]	DOI: 10.2307/1969513
[12]	DOI: 10.1112/blms/25.1.37 · Zbl 0811.20026