INVARIANTS AND EXAMPLES OF GROUP-ACTIONS ON TREES AND LENGTH FUNCTIONS

An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l' to arise from the same action of G on some tree, L = L' and K = K'. Moreover for L non-abelian L = L' implies K = K'. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l' on an H.N.N. group are given, with their associated actions on trees, for which L = L' is abelian but K not-equal K'.