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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $T_{i} T_{i+1}^{-1} T_{i+2} T_{i+3}^{-1} = 1$.


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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: http://dx.doi.org/10.1090/S0002-9947-98-01818-2
PII: S 0002-9947(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