Hilbert space compression and exactness for discrete groups

2004/03/26 by Sarah J. Campbell, Campbell, Sarah J., Graham A. Niblo +1
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Group Theory (math.GR) #math.FA #math.GR

paper · pdf · doi:10.48550/arxiv.math/0403456

14 pages

arxiv created 2004/03/26 · openalex publication_date 2004/03/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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