Retracts of vertex sets of trees and 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 ′. Throughout the article, let G be a group, and let N denote the set of finite cardinals, {0, 1, 2,. . .}. All our G-actions will be on the left. The following extends Definitions II.1.1 of [3] (where A is assumed to have trivial G-action).

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