In [4; 3.1] it is shown that an R-tree T, on which a group G acts, has a unique minimal invariant subtree if there are elements of G with no fixed points. The situation where each element of G fixes some point of T splits into two cases depending on whether the action is bounded or not. In the unbounded case, when T has no invariant subtree, it is shown in Theorem 1 that G is given by an infinite tower of subgroups. In [3] Chiswell constructs an action on a tree that corresponds to a given Lyndon length function defined on G. This construction is used in Theorem 2 to establish a necessary and sufficient condition for two length functions to arise from the same action of G on some tree T, again in the case where each element of G fixes some point of T.

An R-tree T is a non-empty metric space, with metric d, such that there is no subspace homeomorphic to a circle, and for any two points u, u ∈ T there is a unique isometry α : [0, r] → T, with α(0) = u, α(r) = u, where r = d(u, u). It is shown in section 4 of [5] that the completion of an R-tree is again an R-tree. The definition is originally due to Tits [8], where completeness is assumed. In this paper all R-trees will be assumed to be complete. This allows the results of Theorem 1 to be economically expressed.