Warren Dicks and M. J. Dunwoody,
Retracts of vertex sets of trees and the almost stability theorem,
J. Group Theory 10 (2007), 703-721.

Abstract:  Let  G  be a group, let  T  be an (oriented)  G-tree with finite edge stabilizers,
and let  VT  denote the vertex set of  T.  We show that, for each  G-retract  V'
of the  G-set  VT,  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
VT  such that no  G-tree has vertex set  V'.

January 6, 2007 version, 15 pages, available as
 

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