Abstract

We say that a finitely generated group G has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the ℓ 1 -metric.

Highlights

  • We say that a finitely generated group G has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings

  • A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the 1-metric

  • In [DJ99] Dranishnikov and Januszkiewicz show that any Coxeter group admits such an action on a finite product of trees

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Summary

Introduction

We say that a finitely generated group G has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. In view of the lattice example in Sp(n, 1), by Theorem 1.1, having a proper action on a finite product of quasi-trees that gives a quasi-isometric embedding of a group is not enough to expect a proper isometric group action on the Hilbert space. It is unknown if mapping class groups have either property (T) or the Haagerup property. We would like to thank the referee for comments, which improved the presentation of the paper

Separability
Induction
Projection complexes
Projection axioms in δ-hyperbolic spaces
Constants
Preferred axes
Coloring A
Product of quasi-trees X
Curve graphs and subsurface projections
The Masur-Minsky distance formula
Bounded pairs and finiteness
Tight geodesics
Thick distance
Thick distance formula
Separability in the mapping class group
Projection axioms for axes in curve graphs
Full Text
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