Coarse Geometry of Free Products of Metric Spaces

TL;DR

This paper investigates how certain coarse geometric properties are preserved under the free product operation of metric spaces, extending known results from discrete groups to general metric spaces.

Contribution

It establishes that properties like coarse embeddability, Property A, and hyperbolicity are preserved under free products of metric spaces.

Findings

Preservation of coarse embeddability into Hilbert and Banach spaces

Maintenance of Yu's Property A in free products

Hyperbolicity is preserved in free products of metric spaces

Abstract

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.