Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees
Seiten
2011
American Mathematical Society (Verlag)
978-0-8218-4712-1 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-4712-1 (ISBN)
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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.
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.
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.
| Erscheint lt. Verlag | 23.12.2011 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Gewicht | 192 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 0-8218-4712-0 / 0821847120 |
| ISBN-13 | 978-0-8218-4712-1 / 9780821847121 |
| Zustand | Neuware |
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