Number of orbits of branch points of R-trees

An A-tree is a metric space in which any two points are joined by a unique arc. Every arcis isometric to a closed interval of R . When a group G acts on a tree (Z-tree) X without inversion, then X/G is a graph. One gets a presentation of G from the quotient graph X/G with vertex and edge stabilizers attached. For a general R-tree X, there is no such combinatorial structure on X/G . But we can still ask what the maximum number of orbits of branch points of free actions on /{-trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation. © 1993 American Mathematical Society