Real Length Functions in Groups

Abstract

This paper is a study of the structure of a group $G$ equipped with a ’length’ function from $G$ to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon’s work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $|g|$ such that the following axioms are satisfied: $|x| < |xx|$ if $x \ne 1$. $|x| = 0$ if and only if $x = 1$. $|{x^{ - 1}}| = |x|$. $c(x,y) \geq 0$ where $c(x,y) = 1/2(|x| + |y| - |x y^{-1}|)$. $c(x,y) \geq m$ and $c(y,z) \geq m$ imply $c(x,z) \geq m$. In this paper structure theorems are obtained for the cases when $G$ is abelian and when $G$ can be generated by two elements. We first prove that if $G$ is abelian, then $G$ is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of $G$. We prove that if $G$ can be generated by two elements, then $G$ is either free or abelian.