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 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 .

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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