CHARACTERIZATION OF NON-MINIMAL TREE ACTIONS
Lisa A. Carbone, Laura Ciobanu · 2007
Abstract. Let Γ be a group acting without inversions on a tree X. If there is no proper Γ-invariant subtree, we call the action of Γ on X minimal. Here we give a characterization of non-minimal tree actions. For a non-minimal action of Γ on X, wegive structure theorems for the quotient graph of groups for Γ on X, its associated edge-indexed graph and its group of deck transformations. 1. Introduction. Let Γ be a group acting without inversions on a locally finite tree X. Wesay that Γ acts minimally on X if there is no proper Γ-invariant subtree. As is the case with well known results concerning group actions on R-trees, when the hyperbolic length function l(Γ) of Γ is non-zero, there is a unique minimal Γ-invariant subtree X0 ⊂ X (see for example [B]).