## Exactness of one relator groups

HTML articles powered by AMS MathViewer

- by Erik Guentner
- Proc. Amer. Math. Soc.
**130**(2002), 1087-1093 - DOI: https://doi.org/10.1090/S0002-9939-01-06195-0
- Published electronically: October 12, 2001
- PDF | Request permission

## 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 ${\Gamma }=\langle \; X \;|\; R \;\rangle$ 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.## References

- S. Adams,
*Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups*, Topology**33**(1994), no. 4, 765–783. MR**1293309**, DOI 10.1016/0040-9383(94)90007-8 - C. Anantharaman-Delaroche and J. Renault,
*Amenable groupoids*, Monographies de L’Enseignement Math. 36, Geneva, 2000. - Gilbert Baumslag,
*Topics in combinatorial group theory*, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR**1243634**, DOI 10.1007/978-3-0348-8587-4 - Cédric Béguin, Hela Bettaieb, and Alain Valette,
*$K$-theory for $C^\ast$-algebras of one-relator groups*, $K$-Theory**16**(1999), no. 3, 277–298. MR**1681180**, DOI 10.1023/A:1007755408585 - P. Cherix, M. Cowling, P. Jollissaint, P. Julg, and A. Valette,
*Locally compact groups with the Haagerup property*, Unpublished manuscript, 1998. - Alain Connes and Henri Moscovici,
*Cyclic cohomology, the Novikov conjecture and hyperbolic groups*, Topology**29**(1990), no. 3, 345–388. MR**1066176**, DOI 10.1016/0040-9383(90)90003-3 - K. Dykema,
*Exactness of reduced amalgamated free product $C^*$-algebras*, Preprint, 1999. - E. Germain,
*Approximate invariant means for boundary actions of hyperbolic groups*, Appendix to*Amenable Groupoids*[ADR98], 1998. - E. Guentner and J. Kaminker,
*Exactness and the Novikov conjecture*, To appear in Topology, 1999. - E. Guentner and J. Kaminker,
*Addendum to “Exactness and the Novikov conjecture”*, To appear in Topology, 2000. - M. Gromov,
*Spaces and questions*, Unpublished manuscript, 1999. - Nigel Higson and Gennadi 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**1487204**, DOI 10.1090/S1079-6762-97-00038-3 - 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. - G. G. Kasparov,
*Equivariant $KK$-theory and the Novikov conjecture*, Invent. Math.**91**(1988), no. 1, 147–201. MR**918241**, DOI 10.1007/BF01404917 - Eberhard Kirchberg and Simon Wassermann,
*Operations on continuous bundles of $C^*$-algebras*, Math. Ann.**303**(1995), no. 4, 677–697. MR**1359955**, DOI 10.1007/BF01461011 - E. Kirchberg and S. Wassermann,
*Permanence properties of $C^*$-exact groups*, Documenta Mathematica**4**(1999), 513–558. - L. Lance,
*On nuclear $C^*$-algebras*, J. Funct. Anal.**12**(1973), 157–176. - James McCool and Paul E. Schupp,
*On one relator groups and $\textrm {HNN}$ extensions*, J. Austral. Math. Soc.**16**(1973), 249–256. Collection of articles dedicated to the memory of Hanna Neumann, II. MR**0338186**, DOI 10.1017/S1446788700014300 - N. Ozawa,
*Amenable actions and exactness for discrete groups*, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), No. 8, 691–695. - Franz Rádl,
*Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen*, Math. Z.**45**(1939), 429–446 (German). MR**82**, DOI 10.1007/BF01580293 - A. M. Sinclair and R. R. Smith,
*The completely bounded approximation property for discrete crossed products*, Indiana Univ. Math. J.**46**(1997), no. 4, 1311–1322. MR**1631596**, DOI 10.1512/iumj.1997.46.1428 - J. L. Tu,
*Remarks On Yu’s Property A for discrete metric spaces and groups*, Preprint, 2000.

## Bibliographic 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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1873783