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Quasi-actions on trees II: Finite depth Bass-Serre trees
About this Title
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: https://doi.org/10.1090/S0065-9266-2011-00585-X
Published electronically: April 1, 2011
Keywords: must have some keywords
MSC: Primary 20F65
Table of Contents
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
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.- N. Brady and M. R. Bridson, There is only one gap in the isoperimetric spectrum, Geom. Funct. Anal. 10 (2000), no. 5, 1053–1070. MR 1800063, DOI 10.1007/PL00001646
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