FREE PRODUCT WITH AMALGAMATION AND FREE ACTIONS ON R-TREES

The theory of Bass and Serre [17] shows that the actions of a group G on simplicial trees are in natural one-to-one correspondence with splitting of the group G. Group actions on R-trees are natural generalizations of the group actions on simplicial trees (Z-trees). An R-tree is a non-empty metric space X, in which any two points are joined by a unique arc. An R-tree can be infinitely ‘hairy’, and it can ‘branch’ at any points. J. W. Morgan and P. B. Shalen established connections of the theory of group actions on R-trees with hyperbolic geometry and with W. Thurston’s theory of measured laminations. R. Alperin and H. Bass asked the following questions. What group theoretic information about G can be drawn from its action on an R-tree? In particular, how much of Bass-Serre theory goes over for R-tree actions? P. B. Shalen asked the following question. © 1991 Oxford University Press. All rights reserved.