Quasi-actions on trees II: Finite depth Bass-Serre trees

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 G G is a finite graph of coarse Poincaré duality groups, then any finitely generated group quasi-isometric to the fundamental group of G 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 G v G_v which is an n n -dimensional coarse Poincaré duality group for which every incident edge group has positive codimension: the crossing graph of G v G_v is a graph ϵ v ε _v that describes the pattern in which the codimension 1 edge groups incident to G v G_v are crossed by other edge groups incident to G v G_v , and the crossing graph condition requires that ϵ v ε _v be connected or empty.