Small cancellation groups and translation numbers

TL;DR

This paper demonstrates that certain small cancellation groups are translation discrete, provides algorithms for solving power equations within these groups, and shows that their translation numbers are rational with bounded denominators.

Contribution

It establishes translation discreteness for specific small cancellation groups and introduces algorithms for solving power equations and computing translation numbers.

Findings

Groups are translation discrete in the strongest sense

Algorithms for solving x^n=g equations and computing maximum n

Translation numbers are rational with bounded denominators

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 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.