Small cancellation groups and translation numbers

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 g and any n n there is an algorithm deciding whether or not the equation x n = g x^{n}=g has a solution. There is also an algorithm for calculating for each g g the maximum n n such that g g is an n n -th power of some element. We also note that these groups cannot contain isomorphic copies of the group of p p -adic fractions and so in particular of the group of rational numbers. Besides we show that for C ′ ′ ( 4 ) − T ( 4 ) C^{\prime \prime }(4)-T(4) and C ( 3 ) − T ( 6 ) C(3)-T(6) groups all translation numbers are rational and have bounded denominators.