Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.

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 restrictions 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.

In [7] S. Pride gave a family of examples of finitely presented groups of cohomological dimension 2 having no non-trivial action on a simplicial tree. We show here that his examples have no non-trivial action on a Λ-tree, for any ordered abelian group Λ. This provides further slight evidence for an affirmative answer to Question A in §3.1 of [8]. We also give another similar family of examples.

This communication concerns our paper which appeared, without proper proofreading, in Mathematika, 41 (1994), pp. 239–250. Listed below are corrections to those of the misprints/omissions which, in our judgement, most significantly interfere with efficient reading of the paper. We apologize for any inconvenience that may have resulted, and we thank the editors of Mathematika for an opportunity to make up the mistake.