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Exactness of one relator groups


Author: Erik Guentner
Journal: Proc. Amer. Math. Soc. 130 (2002), 1087-1093
MSC (1991): Primary 47L85; Secondary 20E06, 22D15
DOI: https://doi.org/10.1090/S0002-9939-01-06195-0
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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  • [Ada94] S. Adams, Boundary amenability for hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765-783. MR 96g:58104
  • [ADR98] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L'Enseignement Math. 36, Geneva, 2000. CMP 2001:05
  • [Bau93] G. Baumslag, Topics in combinatorial group theory, ETH Lectures in Mathematics, Birkhäuser, Boston, 1993. MR 94j:20034
  • [BBV99] C. Beguin, H. Bettaieb, and A. Valette, $K$-theory for $C^*$-algebras of one-relator groups, $K$-Theory 16 (1999), 277-298. MR 2000c:46133
  • [CCJ+98] P. Cherix, M. Cowling, P. Jollissaint, P. Julg, and A. Valette, Locally compact groups with the Haagerup property, Unpublished manuscript, 1998.
  • [CM90] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture, and hyperbolic groups, Topology 29 (1990), 345-388. MR 92a:58137
  • [Dyk99] K. Dykema, Exactness of reduced amalgamated free product $C^*$-algebras, Preprint, 1999.
  • [Ger98] E. Germain, Approximate invariant means for boundary actions of hyperbolic groups, Appendix to Amenable Groupoids [ADR98], 1998. CMP 2001:05
  • [GK99] E. Guentner and J. Kaminker, Exactness and the Novikov conjecture, To appear in Topology, 1999.
  • [GK00] E. Guentner and J. Kaminker, Addendum to ``Exactness and the Novikov conjecture'', To appear in Topology, 2000.
  • [Gro99] M. Gromov, Spaces and questions, Unpublished manuscript, 1999.
  • [HK97] N. Higson and G. G. Kasparov, Operator $K$-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131-142. MR 99e:46090
  • [HK00] N. Higson and G. G. Kasparov, $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space, To appear in Invent. Math., 2000.
  • [Kas88] G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147-201. MR 88j:58123
  • [KW95] E. Kirchberg and S. Wassermann, Operations on continuous bundles of $C^*$-algebras, Mathematische Annalen 303 (1995), 677-697. MR 96j:46057
  • [KW99] E. Kirchberg and S. Wassermann, Permanence properties of $C^*$-exact groups, Documenta Mathematica 4 (1999), 513-558. CMP 2000:05
  • [Lan73] L. Lance, On nuclear $C^*$-algebras, J. Funct. Anal. 12 (1973), 157-176.
  • [MS73] J. McCool and P. Schupp, On one relator groups and HNN extensions, J. of the Australian Math. Society 16 (1973), 249-256. MR 49:2952
  • [Oza00] N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), No. 8, 691-695. CMP 2000:14
  • [Ser80] J. P. Serre, Trees, Springer, New York, 1980, Translation from French of ``Arbres, Amalgames, $SL_2$'', Astérisque no. 46. MR 82c:20083; MR 57:16426
  • [SS97] A. M. Sinclair and R. R. Smith, The completely bounded approximation property for discrete crossed products, Indiana Univ. Math. J. 46 (1997), 1311-1322. MR 99e:46072
  • [Tu00] J. L. Tu, Remarks On Yu's Property A for discrete metric spaces and groups, Preprint, 2000.

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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: https://doi.org/10.1090/S0002-9939-01-06195-0
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

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