Quasi-actions on trees II: Finite depth Bass-Serre trees

TL;DR

This paper investigates the quasi-isometric rigidity of fundamental groups of finite graphs of coarse Poincare duality groups with finite depth Bass-Serre trees, establishing conditions under which quasi-isometries preserve the graph of groups structure.

Contribution

It proves that under the crossing graph condition, quasi-isometric groups are also fundamental groups of similar graphs of coarse Poincare duality groups, extending rigidity results.

Findings

Quasi-isometries coarsely preserve Bass-Serre tree structures.

Groups satisfying the crossing graph condition are quasi-isometrically rigid.

Main theorem applies to groups with finite depth Bass-Serre trees.

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 Poincare duality groups. The main theorem says that, under certain hypotheses, if G is a finite graph of coarse Poincare duality groups then any finitely generated group quasi-isometric to the fundamental group of 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 preserves 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 G_v which is an n-dimensional coarse Poincare duality group for which every incident edge group has positive codimension: the crossing graph of G_v is a graph e_v that describes the pattern in which the codimension~1 edge groups incident to G_v are crossed by other edge groups incident to G_v, and the crossing graph condition requires that e_v be connected or empty.