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Locally quasiconvex small-cancellation groups
Authors:
Jonathan P. McCammond and Daniel T. Wise
Journal:
Trans. Amer. Math. Soc. 360 (2008), 237-271
MSC (2000):
Primary 20F06, 20F67, 57M07
Posted:
July 23, 2007
MathSciNet review:
2342002
Full-text PDF
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Additional Information
Abstract: In this article we prove several results about the local quasiconvexity behavior of small-cancellation groups. In addition to strengthening our previously obtained positive results, we also describe several families of negative examples. Also, as the strength of the assumed small-cancellation conditions increases, the gap between our positive results and our counterexamples narrows. Finally, as an additional application of these techniques, we include similar results and counterexamples for Coxeter groups.
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- 1.
- J. M. Alonso and et al.
Notes on word hyperbolic groups.
In Group theory from a geometrical viewpoint (Trieste, 1990), pages 3-63. World Sci. Publishing, River Edge, NJ, 1991.
Edited by H. Short. MR 1170363 (93g:57001)
- 2.
- Gilbert Baumslag, W. W. Boone, and B. H. Neumann.
Some unsolvable problems about elements and subgroups of groups.
Math. Scand., 7:191-201, 1959. MR 0163948 (29:1247)
- 3.
- Mladen Bestvina and Noel Brady.
Morse theory and finiteness properties of groups.
Invent. Math., 129(3):445-470, 1997. MR 1465330 (98i:20039)
- 4.
- Robert Bieri, Walter D. Neumann, and Ralph Strebel.
A geometric invariant of discrete groups.
Invent. Math., 90(3):451-477, 1987. MR 914846 (89b:20108)
- 5.
- N. Bourbaki.
Groupes et algèbres de Lie. Chapitres IV-VI.
Hermann, Paris, 1968.
Actualités Scientifiques et Industrielles, No. 1337. MR 0240238 (39:1590)
- 6.
- Noel Brady.
Branched coverings of cubical complexes and subgroups of hyperbolic groups.
J. London Math. Soc. (2), 60(2):461-480, 1999. MR 1724853 (2000j:20076)
- 7.
- Mark Feighn and Michael Handel.
Mapping tori of free group automorphisms are coherent.
Ann. of Math. (2), 149(3):1061-1077, 1999. MR 1709311 (2000i:20050)
- 8.
- S. M. Gersten.
Questions on geometric group theory for the Max Dehn seminar.
Available at ftp.math.utah.edu/u/ma/gersten/MaxDehnSeminar/hyp-quest.ps.
- 9.
- Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev.
Widths of subgroups.
Trans. Amer. Math. Soc., 350(1):321-329, 1998. MR 1389776 (98e:20048)
- 10.
- Catherine Greenhill, Jeong Han Kim, and Nicholas C. Wormald.
Hamiltonian decompositions of random bipartite regular graphs.
J. Combin. Theory Ser. B, 90(2):195-222, 2004. MR 2034027 (2004m:05242)
- 11.
- James E. Humphreys.
Reflection groups and Coxeter groups.
Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
- 12.
- M. Kapovich.
Private communication.
- 13.
- Roger C. Lyndon and Paul E. Schupp.
Combinatorial group theory.
Springer-Verlag, Berlin-New York, 1977.
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064 (58:28182)
- 14.
- J. McCammond and D. Wise.
Coherence, locally quasiconvexity, and the perimeter of  -complexes.
Geom. Funct. Anal. 15:859-927, 2005. MR 2221153
- 15.
- Jonathan P. McCammond and Daniel T. Wise.
Fans and ladders in small cancellation theory.
Proc. London Math. Soc. (3), 84(3):599-644, 2002. MR 1888425 (2003b:20047)
- 16.
- E. Rips.
Subgroups of small cancellation groups.
Bull. London Math. Soc., 14(1):45-47, 1982. MR 642423 (83c:20049)
- 17.
- R. W. Robinson and N. C. Wormald.
Almost all regular graphs are Hamiltonian.
Random Structures Algorithms, 5(2):363-374, 1994. MR 1262985 (95g:05092)
- 18.
- Daniel T. Wise.
Incoherent negatively curved groups.
Proc. Amer. Math. Soc., 126(4):957-964, 1998. MR 1423338 (98f:20016)
- 19.
- Daniel T. Wise.
The residual finiteness of positive one-relator groups.
Comment. Math. Helv., 76(2):314-338, 2001. MR 1839349 (2002d:20043)
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Additional Information
Jonathan P. McCammond
Affiliation:
Department of Mathematics, Universtiy of California, Santa Barbara, Santa Barbara, California 93106
Email:
jon.mccammond@math.ucsb.edu
Daniel T. Wise
Affiliation:
Department of Mathematics, McGill University, Montreal, Quebec, Canada H3A 2K6
Email:
wise@math.mcgill.ca
DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04206-7
PII:
S 0002-9947(07)04206-7
Keywords:
Coherent,
locally quasiconvex
Received by editor(s):
April 26, 2004
Received by editor(s) in revised form:
October 7, 2005
Posted:
July 23, 2007
Additional Notes:
The first author was supported under NSF grants DMS-99781628 and DMS-0101506
The second author was supported by grants from NSERC and NATEQ
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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