Quasi-Actions on Trees II (eBook)

Finite Depth Bass-Serre Trees

Lee Mosher (Autor)

eBook Download: PDF

105 Seiten
American Mathematical Society (Verlag)
978-1-4704-0625-7 (ISBN)

Infos

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 "e;crossing graph condition"e;, 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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 1-4704-0625-X / 147040625X
• ISBN-13: 978-1-4704-0625-7 / 9781470406257