## 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 Q is a finite graph of coarse Poincaré duality groups, then any finitely generated group quasi-isometric to the fundamental group of Q 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 Q _{v} which is an n-dimensional coarse Poincare duality group for which every incident edge group has positive codimension: the crossing graph of Q _{v} is a graph ∈ _{v} that describes the pattern in which the codimension 1 edge groups incident to Q _{v} are crossed by other edge groups incident to Q _{v}, and the crossing graph condition requires that ∈ _{v} be connected or empty.

Original language | English |
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Pages (from-to) | 1-111 |

Number of pages | 111 |

Journal | Memoirs of the American Mathematical Society |

Volume | 214 |

Issue number | 1008 |

DOIs | |

State | Published - Nov 2011 |

## Keywords

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## All Science Journal Classification (ASJC) codes

- Applied Mathematics
- General Mathematics