Memoirs of the American Mathematical Society 2011; 105 pp; softcover Volume: 214 ISBN10: 0821847120 ISBN13: 9780821847121 List Price: US$74 Individual Members: US$44.40 Institutional Members: US$59.20 Order Code: MEMO/214/1008
 This paper addresses questions of quasiisometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre 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 quasiisometric to the fundamental group of \(\mathcal{G}\) is also the fundamental group of a finite graph of coarse Poincaré duality groups, and any quasiisometry between two such groups must coarsely preserve the vertex and edge spaces of their BassSerre 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. Table of Contents  Introduction
 Preliminaries
 Depth zero vertex rigidity
 Finite depth graphs of groups
 Tree rigidity
 Main theorems
 Applications and examples
 Bibliography
 Index
