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A study of exactness for discrete groups

A study of exactness for discrete groups
A study of exactness for discrete groups
We recall the concepts of exactness for both C*-algebras and groups. We explore some new properties linked or equivalent to exactness, including Property A, a second property we term Property O, and Hilbert space compression [GK2, O, Yu]. We use geometric methods to show that a variety of groups satisfy these properties. We then deduce that those groups are exact.

In particular we show that Properties O and A are equivalent. We show that the integers, groups of subexponential growth, amenable groups and free groups satisfy Property O by constructing a family of Ozawa kernels for each case. To construct these families we exploit growth properties of the integers and groups of subexponential growth, Følner’s criterion for amenable groups and geometric properties of the Cayley graph for free groups. For each of these groups we deduce that they are exact and have Property A. Finally we turn to Hilbert space compression to prove our main theorem that groups acting properly and cocompactly on CAT(0) cube complexes are exact and have Property A.

Campbell, Sarah Janet
bad02608-b161-4650-a030-56ebf7356e05
Campbell, Sarah Janet
bad02608-b161-4650-a030-56ebf7356e05
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944

Campbell, Sarah Janet (2005) A study of exactness for discrete groups. University of Southampton, Mathematics, Doctoral Thesis, 164pp.

Record type: Thesis (Doctoral)

Abstract

We recall the concepts of exactness for both C*-algebras and groups. We explore some new properties linked or equivalent to exactness, including Property A, a second property we term Property O, and Hilbert space compression [GK2, O, Yu]. We use geometric methods to show that a variety of groups satisfy these properties. We then deduce that those groups are exact.

In particular we show that Properties O and A are equivalent. We show that the integers, groups of subexponential growth, amenable groups and free groups satisfy Property O by constructing a family of Ozawa kernels for each case. To construct these families we exploit growth properties of the integers and groups of subexponential growth, Følner’s criterion for amenable groups and geometric properties of the Cayley graph for free groups. For each of these groups we deduce that they are exact and have Property A. Finally we turn to Hilbert space compression to prove our main theorem that groups acting properly and cocompactly on CAT(0) cube complexes are exact and have Property A.

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Published date: September 2005
Organisations: University of Southampton, Mathematical Sciences

Identifiers

Local EPrints ID: 361466
URI: http://eprints.soton.ac.uk/id/eprint/361466
PURE UUID: f1e41065-a9fc-4514-ab12-b40cb9c913db
ORCID for Graham Niblo: ORCID iD orcid.org/0000-0003-0648-7027

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Date deposited: 21 Jan 2014 16:14
Last modified: 13 Jun 2019 00:39

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