Coarse Geometry of Free Products of Metric Spaces

Recently, a notion of the free product $X\ast Y$ of two metric spaces $X$ and $Y$ has been introduced by T. Fukaya and T. Matsuka. In this paper, we study coarse geometric permanence properties of the free product $X\ast Y$. We show that if $X$ and $Y$ satisfy any of the following conditions, then $X\ast Y$ 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.