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Finitely approximate groups and actions Part I: The Ribes–Zalesskiĭ property

Published online by Cambridge University Press:  12 March 2014

Christian Rosendal
Christian Rosendal*
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinoisat Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA, E-mail: [email protected], URL: http://www.math.uic.edu/~rosendal
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Abstract

We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

Keywords

Urysohn metric spaceprofinite topologythe Ribes–Zalesskiĭ Theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1297 - 1306
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Ash, C. J., Inevitable graphs: a proof of the type II conjecture and some related decision procedures, International Journal of Algebra and Computation, vol. 1 (1991), pp. 127–146.CrossRefGoogle Scholar
[2]Bhattacharjee, M. and Macpherson, D., A locally finite dense group acting on the random graph, Forum Mathematician, vol. 17 (2005), no. 3, pp. 513–517.Google Scholar
[3]Coulbois, T., Free product, profinite topology and finitely generated subgroups, International Journal of Algebra and Computation, vol. 11 (2001), no. 2, pp. 171–184.CrossRefGoogle Scholar
[4]Hall, M. Jr., Coset representations in free groups, Transactions of the American Mathematical Society, vol. 67 (1949), no. 2, pp. 421–432.CrossRefGoogle Scholar
[5]Hall, M. Jr., A topology for free groups and related groups, Annals of Mathematics, Second Series, vol. 52 (1950), no. 1, pp. 127–139.CrossRefGoogle Scholar
[6]Herwig, B. and Lascar, D., Extending partial automorphisms and the profinite topology on free groups, Transactions of the American Mathematical Society, vol. 352 (2000), no. 5, pp. 1985–2021.CrossRefGoogle Scholar
[7]Hrushovski, E., Extending partial isomorphisms of graphs, Combinatorica, vol. 12 (1992), no. 4, pp. 411–416.CrossRefGoogle Scholar
[8]Pin, J.-E. and Reutenauer, C., A conjecture on the Hall topology for the free group, The Bulletin of the London Mathematical Society, vol. 23 (1991), pp. 356–362.CrossRefGoogle Scholar
[9]Ribes, L. and Zalesskiĭ, P. A., On the profinite topology on a free group, The Bulletin of the London Mathematical Society, vol. 25 (1993), pp. 37–43.CrossRefGoogle Scholar
[10]Rosendal, C., Finitely approximable groups and actions. Part II: Generic representations, this Journal, vol. 76 (2011), no. 4, pp. 1307–1321.Google Scholar
[11]Solecki, S., Extending partial isometries, Israel Journal of Mathematics, vol. 150 (2005), pp. 315–332.CrossRefGoogle Scholar
[12]Uspenskiĭ, V. V., On the group of isometries of the Urysohn universal metric space, Commentationes Mathematicae Universitatis Carotinae, vol. 31 (1990), no. 1, pp. 181–182.Google Scholar