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
Summary
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
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