Quasi-actions on trees: research announcement

TL;DR

This paper introduces new tools for analyzing the quasi-isometric rigidity of group splittings, especially for graphs of coarse Poincaré duality groups, and establishes classification results based on quasi-isometry.

Contribution

It develops techniques for studying quasi-isometric rigidity of graphs of coarse PD groups and proves new rigidity theorems for groups quasi-isometric to such structures.

Findings

Quasi-isometric classification of groups with graphs of coarse PD groups.

Rigidity results for graphs of coarse PD groups with varying dimensions.

Quasi-isometry invariance of fundamental groups of certain group splittings.

Abstract

We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For example, if Gamma is a graph of coarse PD(n) groups for a fixed n, if the Bass-Serre tree of Gamma has infinitely many ends, and if H is a finitely generated group quasi-isometric to pi_1(Gamma), then we prove that H is the fundamental group of a graph of coarse PD(n) groups, with vertex and edge groups quasi-isometric to those of Gamma. We also have quasi-isometric rigidity theorems for graphs of coarse PD groups of nonconstant dimension, under various assumptions on the edge-to-vertex group inclusions.