This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincaré duality groups. The main theorem says that, under certain hypotheses, if \(\mathcal{G}\) is a finite graph of coarse Poincaré duality groups, then any finitely generated group quasi-isometric to the fundamental group of \(\mathcal{G}\) is also the fundamental group of a finite graph of coarse Poincaré duality groups, and any quasi-isometry between two such groups must coarsely preserve the vertex and edge spaces of their Bass-Serre trees of spaces.

Besides some simple normalization hypotheses, the main hypothesis is the “crossing graph condition”, which is imposed on each vertex group \(\mathcal{G}_v\) which is an \(n\)-dimensional coarse Poincaré duality group for which every incident edge group has positive codimension: the crossing graph of \(\mathcal{G}_v\) is a graph \(\epsilon_v\) that describes the pattern in which the codimension 1 edge groups incident to \(\mathcal{G}_v\) are crossed by other edge groups incident to \(\mathcal{G}_v\), and the crossing graph condition requires that \(\epsilon_v\) be connected or empty.