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# memo_has_moved_text();Quasi-actions on trees II: Finite depth Bass-Serre trees

Lee Mosher, Rutgers University, Newark, Michah Sageev, Technion, Israel University of Technology and Kevin Whyte, University of Illinois at Chicago

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 214, Number 1008
ISBNs: 978-0-8218-4712-1 (print); 978-1-4704-0625-7 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00585-X
Published electronically: April 1, 2011
MathSciNet review: 2867450
MSC: Primary 20F65

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Depth zero vertex rigidity
• Chapter 4. Finite depth graphs of groups
• Chapter 5. Tree rigidity
• Chapter 6. Main theorems
• Chapter 7. Applications and examples

### Abstract

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.