Locally quasiconvex small-cancellation groups

McCammond, Jonathan; Wise, Daniel

doi:10.1090/S0002-9947-07-04206-7

Locally quasiconvex small-cancellation groups
HTML articles powered by AMS MathViewer

by Jonathan P. McCammond and Daniel T. Wise PDF

Trans. Amer. Math. Soc. 360 (2008), 237-271 Request permission

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.

References

J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 3–63. Edited by Short. MR 1170363
Gilbert Baumslag, W. W. Boone, and B. H. Neumann, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7 (1959), 191–201. MR 163948, DOI 10.7146/math.scand.a-10572
Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330, DOI 10.1007/s002220050168
Robert Bieri, Walter D. Neumann, and Ralph Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), no. 3, 451–477. MR 914846, DOI 10.1007/BF01389175
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Noel Brady, Branched coverings of cubical complexes and subgroups of hyperbolic groups, J. London Math. Soc. (2) 60 (1999), no. 2, 461–480. MR 1724853, DOI 10.1112/S0024610799007644
Mark Feighn and Michael Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math. (2) 149 (1999), no. 3, 1061–1077. MR 1709311, DOI 10.2307/121081
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.
Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998), no. 1, 321–329. MR 1389776, DOI 10.1090/S0002-9947-98-01792-9
Catherine Greenhill, Jeong Han Kim, and Nicholas C. Wormald, Hamiltonian decompositions of random bipartite regular graphs, J. Combin. Theory Ser. B 90 (2004), no. 2, 195–222. MR 2034027, DOI 10.1016/j.jctb.2003.07.001
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
M. Kapovich. Private communication.
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
J. P. McCammond and D. T. Wise, Coherence, local quasiconvexity, and the perimeter of 2-complexes, Geom. Funct. Anal. 15 (2005), no. 4, 859–927. MR 2221153, DOI 10.1007/s00039-005-0525-8
Jonathan P. McCammond and Daniel T. Wise, Fans and ladders in small cancellation theory, Proc. London Math. Soc. (3) 84 (2002), no. 3, 599–644. MR 1888425, DOI 10.1112/S0024611502013424
E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR 642423, DOI 10.1112/blms/14.1.45
R. W. Robinson and N. C. Wormald, Almost all regular graphs are Hamiltonian, Random Structures Algorithms 5 (1994), no. 2, 363–374. MR 1262985, DOI 10.1002/rsa.3240050209
Daniel T. Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998), no. 4, 957–964. MR 1423338, DOI 10.1090/S0002-9939-98-04146-X
Daniel T. Wise, The residual finiteness of positive one-relator groups, Comment. Math. Helv. 76 (2001), no. 2, 314–338. MR 1839349, DOI 10.1007/PL00000381