INHERITED GROUP ACTIONS ON [R-TREES ZAD KHAN AND DAVID L. WILKENS Introduction. Group actions on R-trees may be split into different types, and in Section 1 of this paper five distinct types are defined, with one type splitting into two sub-types. For a group G acting as a group of isometries on an R-tree, conditions are considered under which a subgroup or a factor group may inherit the same type of action as G. In Section 2 subgroups of finite index are considered, and in Section 3 normal subgroups and also factor groups are considered. The results obtained here, Theorems 2.1 and 3.4, allow restric- tions on possible types of actions for hypercentral, hypercyclic and hyperabelian groups to be given in Theorem 3.6. In Section 4 finitely generated subgroups are considered, and this gives rise to restrictions on possible actions for groups with certain local properties. The results throughout are stated in terms of group actions on trees. Using Chiswell's construction in [3], they could equally be stated in terms of restrictions on possible types of Lyndon length functions. §1. Types of actions. Let a group G act as a group of isometries on an R- tree T. The translation length (or hyperbolic length) function L: G->U is defined by L(x) = inf {d(u, xu); ueT}, where d is the metric on T. The infimum is attained for each xeG. An element x is elliptic (or non-Archimedean) if L(x) = 0, which occurs if, and only if, x fixes some point of T. The subset of elliptic elements of G is denoted by N. An element x that does not fix any point of T is said to be hyperbolic (or Archimedean). For xe G, either elliptic or hyperbolic, the characteristic set C x = {we T; L(x) = d(u, xu)}, which is a subtree of T. So for xeN, C x is the subtree of points fixed by x. For x$N, C x is an isometric copy of R on which x acts as a translation by distance L(x), and is said to be the axis for x in T. For r # 0, the element x r translates by a distance rL(x) on the same axis as x. So C> = C x , and the order of a hyperbolic element x is necessarily infinite. Lyndon length functions were introduced in [7], and Chiswell showed in [3] that a Lyndon length function /„ : G-*U is defined by l u (x) = d(u, xu), for each point ueT. Conversely given a Lyndon length function /: G-»R, Chiswell gives a construction for an R-tree T, an action of G on T, and a point ueT, such that /= /„. The translation length function is given in terms of the Lyndon length functions by the identity L(x) = max (/„(x 2 ) - l v (x), 0), for any veT. An action is said to be bounded if the Lyndon length function /„ is bounded for some (and so any) veT. Reference for the definitions and properties above may be made to Culler and Morgan [4], or Alperin and Bass [1], where actions on more general A- trees are considered (A being an ordered abelian group). However, readers [MATHEMATIKA, 42 (1995), 206-213]