Trees and Length Functions on Groups

Abstract

Length functions on groups were introduced by Lyndon to axiomatize cancellation arguments used in the proofs of subgroup theorems for free groups and free products. To every group with a length function we associate a metric which can be imbedded into a tree on which the group acts as a group of automorphisms. The original length function can then be recovered from the action of the group on the tree. With the aid of results of Tits on automorphism groups of trees it is possible to give simpler proofs of many results on length functions. Further, we characterize groups with length functions whose set of non-Archimedean elements is an unbounded group, thereby extending results of Wilkens and Hoare [14], (6) on length functions and normal subgroups. We also characterize the case in which the set of Archimedean elements is a group.