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Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees
Lee Mosher
Michah Sageev
Kevin Whyte
其他書名
Finite Depth Bass-Serre Trees
出版
American Mathematical Soc.
, 2011
主題
Mathematics / Geometry / General
Mathematics / Geometry / Algebraic
Mathematics / Group Theory
ISBN
0821847120
9780821847121
URL
http://books.google.com.hk/books?id=KBq6AwAAQBAJ&hl=&source=gbs_api
EBook
SAMPLE
註釋
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 Poincare duality groups. The main theorem says that, under certain hypotheses, if $\mathcal{G}$ is a finite graph of coarse Poincare 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 Poincare 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 Poincare 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.