Coarse geometry of free products of metric spaces

Abstract

Recently, a notion of the free product $X⁎Y$ of two metric spaces X and Y has been introduced by T. Fukaya and T. Matsuka in their study of the coarse Baum-Connes conjecture. In this paper, we study coarse geometric permanence properties of the free product $X⁎Y$. We show that if X and Y satisfy any of the following conditions, then $X⁎Y$ also satisfies that condition: (1) they are coarsely embeddable into a Hilbert space or a uniformly convex Banach space; (2) they have Yu's Property A; (3) they are hyperbolic spaces. These generalize the corresponding results for discrete groups to the case of metric spaces.