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

Lee Mosher Rutgers University, Newark, Newark, NJ
Michah Sageev Technion, Israel University of Technology, Haifa, Israel
Kevin Whyte University of Illinois at Chicago, Chicago, IL
Available Formats:
Electronic ISBN: 978-1-4704-0625-7
Product Code: MEMO/214/1008.E
List Price: $74.00 MAA Member Price:$66.60
AMS Member Price: $44.40 Click above image for expanded view Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees Lee Mosher Rutgers University, Newark, Newark, NJ Michah Sageev Technion, Israel University of Technology, Haifa, Israel Kevin Whyte University of Illinois at Chicago, Chicago, IL Available Formats:  Electronic ISBN: 978-1-4704-0625-7 Product Code: MEMO/214/1008.E  List Price:$74.00 MAA Member Price: $66.60 AMS Member Price:$44.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 2142011; 105 pp
MSC: Primary 20;

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.

• Chapters
• 1. Introduction
• 2. Preliminaries
• 3. Depth Zero Vertex Rigidity
• 4. Finite Depth Graphs of Groups
• 5. Tree Rigidity
• 6. Main Theorems
• 7. Applications and Examples
• Requests

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Volume: 2142011; 105 pp
MSC: Primary 20;

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.

• Chapters
• 1. Introduction
• 2. Preliminaries
• 3. Depth Zero Vertex Rigidity
• 4. Finite Depth Graphs of Groups
• 5. Tree Rigidity
• 6. Main Theorems
• 7. Applications and Examples
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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