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Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees
Lee Mosher, Rutgers University, Newark, NJ, Michah Sageev, Technion, Israel University of Technology, Haifa, Israel, and Kevin Whyte, University of Illinois at Chicago, IL

Memoirs of the American Mathematical Society
2011; 105 pp; softcover
Volume: 214
ISBN-10: 0-8218-4712-0
ISBN-13: 978-0-8218-4712-1
List Price: US$74
Individual Members: US$44.40
Institutional Members: US$59.20
Order Code: MEMO/214/1008
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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.

Table of Contents

  • Introduction
  • Preliminaries
  • Depth zero vertex rigidity
  • Finite depth graphs of groups
  • Tree rigidity
  • Main theorems
  • Applications and examples
  • Bibliography
  • Index
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