A cohomological characterisation of Yu’s property A for metric spaces

Brodzki, Jacek¹ ; Niblo, Graham¹ ; Wright, Nick¹

¹ School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1SH, England

Geometry & topology, Tome 16 (2012) no. 1, pp. 391-432.

Résumé

We develop a new framework for cohomology of discrete metric spaces and groups which simultaneously generalises group cohomology, Roe’s coarse cohomology, Gersten’s ${\ell }^{\infty }$–cohomology and Johnson’s bounded cohomology. In this framework we give an answer to Higson’s question concerning the existence of a cohomological characterisation of Yu’s property A, analogous to Johnson’s characterisation of amenability. In particular, we introduce an analogue of invariant mean for metric spaces with property A. As an application we extend Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This provides an alternative proof of Nowak’s result that the infinite dimensional cube does not have property A.

Détail

DOI : 10.2140/gt.2012.16.391

Keywords: Property A, bounded cohomology, coarse geometry, group cohomology

Bibliographie

[1] J Brodzki, G A Niblo, P W Nowak, N J Wright, Amenable actions, invariant means and bounded cohomology

[2] J Brodzki, G A Niblo, P W Nowak, N J Wright, A homological characterization of topological amenability

[3] J Brodzki, G A Niblo, N J Wright, Pairings, duality, amenability, and bounded cohomology, to appear in J. Eur. Math. Soc. (JEMS)

[4] R G Douglas, P W Nowak, Invariant expectations and vanishing of bounded cohomology for exact groups, J. Topol. Anal. 3 (2011) 89

[5] S M Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (1998) 1031

[6] N Higson, J Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000) 143

[7] B E Johnson, Cohomology of Banach Algebras, Mem. Amer. Math. Soc. 127, American Mathematical Society (1972)

[8] A Khukhro, N J Wright, Expanders and property A, Algebr. Geom. Topol. 12 (2012) 37

[9] P W Nowak, Coarsely embeddable metric spaces without Property A, J. Funct. Anal. 252 (2007) 126

[10] N Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 691

[11] H Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford (1968)

[12] J Roe, Exotic cohomology and index theory, Bull. Amer. Math. Soc. 23 (1990) 447

[13] J Roe, Lectures on coarse geometry, University Lecture Series 31, American Mathematical Society (2003)

[14] J Von Neumann, Zur allgemeinen Theorie des Ma\sses, Fund. Math. 13 (1929) 73

[15] G Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000) 201