Some refinements of the almost stability theorem

Let G be a group, let T be an (oriented) G-tree with finite edge stabilizers, and let V T denote the vertex set of T . We show that, for each G-retract V ′ of the G-set V T , there exists a G-tree whose edge stabilizers are finite and whose vertex set is V ′. This fact leads to various new consequences of the almost stability theorem. We also give an example of a group G, a G-tree T and a G-retract V ′ of V T such that no G-tree has vertex set V ′. 2000 Mathematics Subject Classification. Primary: 20E08; Secondary: 05C25, 20J05.