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Quasi-actions on trees. II: Finite depth Bass-Serre trees. (English) Zbl 1234.20034

Mem. Am. Math. Soc. 1008, v, 105 p. (2011).
Summary: 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 \(\varepsilon_v\) 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 \(\varepsilon_v\) be connected or empty.
For part I cf. the authors [Ann. Math. (2) 158, No. 1, 115-164 (2003; Zbl 1038.20016)].

MSC:

20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
57M07 Topological methods in group theory

Citations:

Zbl 1038.20016
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References:

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