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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Exactness of one relator groups


Author: Erik Guentner
Journal: Proc. Amer. Math. Soc. 130 (2002), 1087-1093
MSC (1991): Primary 47L85; Secondary 20E06, 22D15
Published electronically: October 12, 2001
MathSciNet review: 1873783
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Abstract: A discrete group ${\Gamma}$ is $C^*$-exact if the reduced crossed product with ${\Gamma}$ converts a short exact sequence of ${\Gamma}$-$C^*$-algebras into a short exact sequence of $C^*$-algebras. A one relator group is a discrete group ${\Gamma}$ admitting a presentation ${\Gamma}=\langle\; X \;\vert\; R \;\rangle$ where $X$ is a countable set and $R$ is a single word over $X$. In this short paper we prove that all one relator discrete groups are $C^*$-exact. Using the Bass-Serre theory we also prove that a countable discrete group $\Gamma$ acting without inversion on a tree is $C^*$-exact if the vertex stabilizers of the action are $C^*$-exact.


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Additional Information

Erik Guentner
Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, Indiana 46202-3216
Address at time of publication: Mathematical Sciences Research Institute, 100 Centennial Drive, #5070, Berkeley, California 94702-5070
Email: guentner@msri.org

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06195-0
PII: S 0002-9939(01)06195-0
Keywords: Group $C^*$-algebra, $C^*$-exactness
Received by editor(s): October 9, 2000
Published electronically: October 12, 2001
Additional Notes: The author was supported with funds from the NSF
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society