COVERING THEORY FOR GRAPHS OF GROUPS

A tree action (G, X), consisting of a group G acting on a tree X, is e ncoded by a 'quotient graph of groups' A = G\\X. We introduce here the appropriate notion of morphism A --> A' = G'\\X', that encodes a morp hism (G, X) --> (G', X') of tree actions. In particular, we characteri ze 'coverings' G\\X --> G'\\X corresponding to inclusions of subgroups G less-than-or-equal-to G'. This is a useful tool for producing subgr oups of G with prescribed properties. It also yields a strong Conjugac y Theorem for groups acting freely on X. We also prove the following m ild generalizations of theorems of Howie and Greenberg. Suppose that G acts discretely on X, i.e. that each vertex stabilizer is finite. Let H and K be finitely generated subgroups of G. Then H and K is finitel y generated. If H and K are commensurable, then H (and K) have finite index in [H, K].