Coarse Geometry of Free Products of Metric Spaces

TL;DR

This work investigates the persistence of key coarse geometric properties under the free product operation for metric spaces. It develops explicit embedding constructions that transfer coarse embeddability from factors to the free product, and extends these results from Hilbert spaces to uniformly convex Banach spaces using Day's theorem and common moduli of convexity. It also shows that Yu's Property A and Gromov hyperbolicity are preserved under free products, employing a Bass-Serre tree framework to unify the proofs. The results extend known group-theoretic permanence properties to the setting of general metric spaces, with implications for coarse Baum-Connes type questions and large-scale geometry.

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.

Figures (12)

• Figure 4: Components of the Free Product $X\ast Y$
• Figure 8: Visualization of functions $\rho'_1(t)$ (red, left) and $\rho_1(t)$ (blue, right). The function $\rho_1(t)$ is constructed to satisfy condition (3) in Lemma \ref{['normalization']}.
• Figure 9: Conceptual illustration of the construction of the coarse embedding $F: X \ast Y \to \mathcal{H}_{X\ast Y}$.
• Figure 10: Case $1$ configuration: Points $(\omega, z, t)$ and $(\omega', z', t')$ in $X \ast Y$ represented by words with a common prefix $\rho$ but ending in different types of components. Here, $\omega = \rho x_1 y_1 \dots x_m y_m$ and $\omega' = \rho x'_1 y'_1 \dots x'_n y'_n x'_{n+1}$.
• Figure 11: Case $2$ configuration: Points $(\omega, z, t)$ and $(\omega', z', t')$ in $X \ast Y$ represented by words with different starting types. Here, $\omega=x_{1}y_{1}\dots x_{m}y_{m}$ and $\omega'=y'_{1}x'_{2}y'_{2}\dots x'_{n}y'_{n}$.

Theorems & Definitions (9)

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