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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small cancellation groups and
translation numbers

Author: Ilya Kapovich
Journal: Trans. Amer. Math. Soc. 349 (1997), 1851-1875
MSC (1991): Primary 20F06; Secondary 20F10, 20F32
MathSciNet review: 1357396
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Abstract: In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any $g$ and any $n$ there is an algorithm deciding whether or not the equation $x^{n}=g$ has a solution. There is also an algorithm for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for $C^{\prime \prime }(4)-T(4)$ and $C''(3)-T(6)$ groups all translation numbers are rational and have bounded denominators.

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Additional Information

Ilya Kapovich
Affiliation: Department of Mathematics, Graduate School and University Center of the City University of New York, 33 West 42nd Street, New York, New York 10036
Address at time of publication: Department of Mathematics, Hill Center, Busch Campus, Rutgers University at New Brunswick, Piscataway, New Jersey 08854

Received by editor(s): May 26, 1994
Received by editor(s) in revised form: October 30, 1995
Additional Notes: This research was supported by the Robert E. Gilleece Fellowship at the CUNY Graduate Center.
Article copyright: © Copyright 1997 American Mathematical Society