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On the hyperbolicity of small cancellation groups and one-relator groups


Authors: S. V. Ivanov and P. E. Schupp
Journal: Trans. Amer. Math. Soc. 350 (1998), 1851-1894
MSC (1991): Primary 20F05, 20F06, 20F32
DOI: https://doi.org/10.1090/S0002-9947-98-01818-2
MathSciNet review: 1401522
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Abstract: In the article, a result relating to maps (= finite planar connected and simply connected 2-complexes) that satisfy a $C(p)\&T(q)$ condition (where $(p,q)$ is one of $(3,6)$, $(4,4)$, $(6,3)$ which correspond to regular tessellations of the plane by triangles, squares, hexagons, respectively) is proven. On the base of this result a criterion for the Gromov hyperbolicity of finitely presented small cancellation groups satisfying non-metric $C(p)\&T(q)$-conditions is obtained and a complete (and explicit) description of hyperbolic groups in some classes of one-relator groups is given: All one-relator hyperbolic groups with $> 0$ and $\le 3$ occurrences of a letter are indicated; it is shown that a finitely generated one-relator group $G$ whose reduced relator $R$ is of the form $R \equiv a T_{0} a T_{1} \dots a T_{n-1}$, where the words $T_{i}$ are distinct and have no occurrences of the letter $a^{\pm 1}$, is not hyperbolic if and only if one has in the free group that (1) $n=2$ and $T_{0} T_{1}^{-1}$ is a proper power; (2) $n = 3$ and for some $i$ it is true (with subscripts $\operatorname{mod} 3$) that $T_{i} T_{i+1}^{-1} T_{i} T_{i+2}^{-1} = 1$; (3) $n = 4$ and for some $i$ it is true (with subscripts $\operatorname{mod} 4$) that $T_{i} T_{i+1}^{-1} T_{i+2} T_{i+3}^{-1} = 1$.


References [Enhancements On Off] (What's this?)

  • [BF] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, Jour. Diff. Geometry 35(1992), 85-101. MR 93d:53053
  • [B] M. Bridson, On the existence of flat planes in spaces of non-positive curvature, Proc. of the AMS 123(1995), 223-235. MR 95d:53048
  • [ECHLPT] D.B.A. Epstein with J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, Word processing in groups, Jones and Bartlett Publishers, 1992. MR 93i:20036
  • [GS1] S. M. Gersten and H. Short, Small cancellation theory and automatic groups, Invent. Math. 102(1990), 305-334. MR 92c:20058
  • [GS2] S. M. Gersten and H. Short, Small cancellation theory and automatic groups: Part II, Invent. Math. 105(1991), 641-662. MR 92j:20030
  • [GH] E. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d'aprés Mikhael Gromov, Birkhaüser, 1990. MR 92f:53050
  • [Gr] M. Gromov, Hyperbolic groups, in Essays in Group Theory, ed. S.M. Gersten, M.S.R.I. Pub. 8, Springer, 1987, 75-263. MR 89e:20070
  • [KM] O. Kharlampovich and A. Myasnikov, Hyperbolic groups and free constructions, Trans. AMS 350 (1998), 571-613.
  • [LS] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer-Verlag, 1977. MR 58:28182
  • [MKS] W. Magnus, J. Karrass, and D. Solitar, Combinatorial group theory, Interscience Pub., John Wiley and Sons, 1966. MR 34:7617
  • [MO] K.V. Mikhajlovskii and A.Yu. Ol'shanskii, Some constructions relating to hyperbolic groups, preprint, Moscow State University, 1994.
  • [Ol] A. Yu. Ol'shanskii, Geometry of defining relations in groups, Nauka, Moscow, 1989; English translation, Math. and Its Applications (Soviet series), 70 Kluwer Acad. Publishers, 1991. MR 91i:20035; MR 93g:20071

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Additional Information

S. V. Ivanov
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: ivanov@math.uiuc.edu

P. E. Schupp
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: schupp@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01818-2
Received by editor(s): May 13, 1995
Received by editor(s) in revised form: May 15, 1996
Additional Notes: The first author is supported in part by an Alfred P. Sloan Research Fellowship, a Beckman Fellowship, and NSF grant DMS 95-01056.
Article copyright: © Copyright 1998 American Mathematical Society

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