Covering theory for graphs of groups

AbstractA tree action (G, X), consisting of a group G acting on a tree X, is encoded by a ‘quotient graph of groups’ A=G⧹⧹X. We introduce here the appropriate notion of morphism A→A′= G′⧹⧹X′, that encodes a morphism (G, X)→(G′, X′) of tree actions. In particular, we characterize ‘coverings’ G⧹⧹X→G′⧹⧹X corresponding to inclusions of subgroups G⩽G′. This is a useful tool for producing subgroups of G with prescribed properties. It also yields a strong Conjugacy Theorem for groups acting freely on X.We also prove the following mild 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∩K is finitely generated. If H and K are commensurable, then H (and K) have finite index in 〈H, K〉