Small cancellation groups and translation numbers

Details Small cancellation groups and translation numbers Small cancellation groups and translation numbers

1996-11-04

arxiv.org/abs/math/9611203v1

Mathematics

Group Theory

Post-Script file only, 22 pages. To appear in Trans. Amer. Math. Soc

Scientific paper

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 dis crete 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 gr oup of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for C''(4)-T(4) and C''(3)-T(6) groups all translation numbers are rational and have bounded denominators.

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