Remarks on Yu’s ‘property A’ for discrete metric spaces and groups

Abstract

Abstract. -Guoliang Yu has introduced a property on discrete metric spaces and groups, which is a weak form of amenability and which has important applications to the Novikov conjecture and the coarse Baum-Connes conjecture. The aim of the present paper is to prove that property in particular examples, like spaces with subexponential growth, amalgamated free products of discrete groups having property A and HNN extensions of discrete groups having property A. Résumé (Remarques sur la propriété A de Yu pour les espaces métriques et les groupes discrets)Guoliang Yu a introduit une propriété sur les espaces métriques et les groupes discrets, qui est une forme faible de moyennabilité et qui a d'importantes applications a la conjecture de Novikov et la conjecture de Baum-Connes "coarse". Le but de cet article est de démontrer cette propriété dans des cas particuliers, tels que les espacesà croissance sous-exponentielle, les produits libres amalgamés de groupes discrets ayant la propriété A et les extensions HNN de groupes discrets ayant la propriété A.

Asymptotic properties of groups acting on complexes

“…Tu proved, [15], that the fundamental group of a finite graph of groups in which each vertex group has property A will have property A. In [1], the author generalized Tu's results to groups acting by isometries on metric spaces with finite asdim .…”

“…Higson and Roe [12] showed that finitely generated groups with finite asymptotic dimension have property A. In [15], J.-L. Tu proved that property A is preserved by the fundamental group of a finite graph of groups where the vertex groups all have property A. At first it was not known whether there could be a finitely generated group not having property A.…”

Constructions preserving Hilbert space uniform embeddability of discrete groups

“…On the other hand, the corresponding result for free products with amalgam is considerably more difficult, in view of the fact that the common subgroup of the amalgam can introduce considerable distortion into the product. Our proof is based on a suitable adaptation of an argument given by Tu in his work on Property A [19]; although we are not able to verify a number of assertions concerning the metric defined in Section 9 in Tu's paper, we are able to adapt his arguments to the present context. The proof we give works equally well for countable exact groups (see Proposition 6.8 and Theorem 6.9), and is unrelated to Dykema's original proof that the class of countable exact groups is closed under free products with amalgam [9], [8].…”

“…Nevertheless, for our X Γ these inclusions are isometries, a fact that may be traced back to the manner in which the vertex and edge spaces of X Γ are metrized as subspaces of Γ, which circumvents any distortion that may have otherwise been introduced by the amalgamating subgroup. The following proposition has no analog in Tu's work [19]; nevertheless, we require it in order to complete our arguments.…”

“…We now turn attention to the proof of that theorem. We suitably adapt the method employed by Tu in his study of Property A for discrete metric spaces [19].…”