# Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees

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*Lee Mosher; Michah Sageev; Kevin Whyte*

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

# Table of Contents

## Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees

- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 1118
- Chapter 3. Depth Zero Vertex Rigidity 2936
- Chapter 4. Finite Depth Graphs of Groups 4956
- Chapter 5. Tree Rigidity 5966
- Chapter 6. Main Theorems 8794
- Chapter 7. Applications and Examples 8996
- Bibliography 101108
- Index 105112 free