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 condition (where is one of , , 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 -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 and occurrences of a letter are indicated; it is shown that a finitely generated one-relator group whose reduced relator is of the form , where the words are distinct and have no occurrences of the letter , is not hyperbolic if and only if one has in the free group that (1) and is a proper power; (2) and for some it is true (with subscripts ) that ; (3) and for some it is true (with subscripts ) that .

**[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