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Cuntz—Krieger algebras associated with Fuchsian groups

Published online by Cambridge University Press:  19 September 2008

J. S. Spielberg
J. S. Spielberg
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA
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Abstract

For Fuchsian groups of the first kind containing parabolic elements, it is shown that the action on a suitable disconnection of the limit circle generates a Cuntz—Krieger C*-algebra. This clarifies and generalizes the situation of the subalgebra within O2, and provides a new proof of the simplicity and nuclearity of certain Cuntz—Krieger algebras. The proof relies on the Markov partition obtained from a suitable fundamental polygon for the group. Counter examples are given if an unsuitable fundamental polygon is used.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 3 , September 1993 , pp. 581 - 595
Copyright
Copyright © Cambridge University Press 1993

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References

REFERENCES

[Ana]Anantharaman-Delaroche, C.. Systèmes dynamiques non commutatifs et moennabilité. Math. Ann. 279 (1987), 297–315.CrossRefGoogle Scholar
[AS]Archbold, R. J. & Spielberg, J.. Topologically free actions and ideals in discrete C*-dynamical systems. Proc. Edinburgh Math. Soc., to appear.Google Scholar
[Bea]Beardon, A. F.. The Geometry of Discrete Groups. Springer: New York, 1983.CrossRefGoogle Scholar
[BS]Bowen, R. & Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. IHES 50 (1979), 153–179.CrossRefGoogle Scholar
[Cho]Choi, M.-D.. A simple C*-algebra generated by two finite order unitaries. Canad. J. Math. 31 (1979), 867–880.CrossRefGoogle Scholar
[Cun]Cuntz, J.. Simple C*-algebras generated by isometries. Commun. Math. Phys. 57 (1977), 173–185.CrossRefGoogle Scholar
[CK]Cuntz, J. & Krieger, W.. A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
[Kee]Keen, L.. Canonical polygons for finitely generated Fuchsian groups. Acta Math. 115 (1966), 1–16.CrossRefGoogle Scholar
[Mas]Maskit, B.. On Poincaré's theorem for fundamental polygons. Adv. Math. 7 (1971), 219–230.CrossRefGoogle Scholar
[QS]Quigg, J. C. & Spielberg, J.. Regularity and hyporegularity in C*-dynamical systems. Houston J. Math. 18 (1992), 139–152.Google Scholar
[Ser1]Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. & Dynam. Sys. 1 (1981), 337–360.CrossRefGoogle Scholar
[Ser2]Series, C.. Non-Euclidean geometry, continued fractions, and ergodic theory. Math. Intelligencer 4 (1982), 24–31.CrossRefGoogle Scholar
[Spi]Spielberg, J.. Free-product groups, Cuntz-Krieger algebras, and covariant maps. Int. J. Math. 2 (1991), 457–476.CrossRefGoogle Scholar