[Paper Review] Hilbert space compression and exactness for discrete groups

Sarah J. Campbell, Graham A. Niblo|ArXiv.org|Mar 26, 2004

Geometric and Algebraic Topology11 references4 citations

TL;DR

This paper establishes that the Hilbert space compression of any finite-dimensional CAT(0) cube complex is 1, by constructing a family of large-scale Lipschitz embeddings into a Hilbert space with asymptotic compression arbitrarily close to 1. As a consequence, any discrete group acting properly and co-compactly on such a complex has Hilbert space compression 1 and is therefore exact, extending exactness to a broad class of groups including free groups, Coxeter groups, right-angled Artin groups, and small cancellation groups.

ABSTRACT

We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.

Motivation & Objective

• To determine the Hilbert space compression of finite-dimensional CAT(0) cube complexes.
• To establish exactness for discrete groups acting properly and co-compactly on CAT(0) cube complexes.
• To extend the class of exact groups beyond those with finite asymptotic dimension or amenability.
• To generalize the Guentner-Kaminker result on Hilbert space compression and exactness to a broader class of groups via cube complex geometry.

Proposed method

• Construct a family of embeddings $ f_\epsilon $ from the vertex set of a CAT(0) cube complex into $ \ell^2(H, \mathbb{R}) $, the Hilbert space of square-summable functions on the set of hyperplanes.
• Use normal cube paths—canonical geodesics in CAT(0) cube complexes—to define the embedding and ensure large-scale Lipschitz continuity.
• Define weights $ w_i = i^{2\epsilon} $ for hyperplanes crossed along paths, and use them to assign coordinates in the Hilbert space.
• Prove that each $ f_\epsilon $ has asymptotic compression at least $ 1/2 + \epsilon $ by comparing distances in the complex to their images in the Hilbert space.
• Leverage the quasi-isometry invariance of Hilbert space compression to transfer the result from the vertex set to the full complex and to the group.
• Apply the Guentner-Kaminker theorem, which states that groups with Hilbert space compression > 1/2 are exact, to conclude exactness for the group.

Experimental results

Research questions

• RQ1 What is the Hilbert space compression of a finite-dimensional CAT(0) cube complex?
• RQ2 Can the Guentner-Kaminker criterion for exactness be extended to groups acting on CAT(0) cube complexes?
• RQ3 How can the construction of embeddings with high asymptotic compression be adapted from trees to higher-dimensional CAT(0) cube complexes?
• RQ4 What role do normal cube paths play in constructing embeddings with controlled distortion?
• RQ5 Which classes of groups can be shown to be exact using this Hilbert space compression approach?

Key findings

• The Hilbert space compression of any finite-dimensional CAT(0) cube complex is exactly 1.
• For each $ \epsilon \in (0, 1/2) $, there exists a large-scale Lipschitz embedding $ f_\epsilon $ of the vertex set into $ \ell^2(H, \mathbb{R}) $ with asymptotic compression at least $ 1/2 + \epsilon $.
• The group $ G $, when acting properly and co-compactly on a CAT(0) cube complex, has Hilbert space compression 1, as it is quasi-isometric to the complex.
• The class of groups covered by this result includes free groups, finitely generated Coxeter groups, right-angled Artin groups, and all word-hyperbolic groups satisfying the B(6) condition.
• All finitely presented groups satisfying the B(4)-T(4) small cancellation condition are exact, as they act properly and co-compactly on CAT(0) cube complexes.
• The fundamental groups of 3-manifolds admitting CAT(0) cube complex structures are also exact, including those arising from canonical surgeries defined by link diagrams.