Free products of coarsely convex spaces and the coarse Baum-Connes conjecture

Abstract

The first author and Oguni introduced a wide class of metric spaces, called coarsely convex spaces. It includes Gromov hyperbolic metric spaces, CAT(0) spaces, systolic complexes, proper injective metric spaces. We introduce the notion of free products of metric spaces and show that free products of symmetric geodesic coarsely convex spaces are also symmetric geodesic coarsely convex spaces. As an application, it follows that free products of symmetric geodesic coarsely convex spaces satisfy the coarse Baum-Connes conjecture.

Figures (8)

• Figure 1: The free product $X*Y$. Here, $x_0,x_1^{\prime} \in X_0^*$ and $y_0, y_0^{\prime} \in Y_0^*$.
• Figure 3: An example of case \ref{['item:w-sub-t']}. Let $p=(\omega,u)$ and $q=(\omega x_0^{\prime}y_0^{\prime}x_1^{\prime},v)$, where $\omega \in W_X \setminus \{\epsilon\}$, $x_i^{\prime} \in X_0^*$, $y_0^{\prime} \in Y_0^*$, $u \in X$, and $v \in Y$.
• Figure 4: An example of \ref{['III']}. Let $p=(\rho x_0y_0, u)$ and $q=(\rho x^{\prime}_0y^{\prime}_0x^{\prime}_1, v)$, where $x_i,x_i^{\prime} \in X_0^*$, $y_0, y_0^{\prime} \in Y_0^*$, $u \in X$, and $v \in Y$. Here, $\rho \in W_X$ and $x_0 \neq x_0^{\prime}$.
• Figure 7: An example of $\Gamma(a,b) \in \mathcal{L}_*$
• Figure 8: An example of geodesic triangles of $X*Y$.

Theorems & Definitions (5)

• proof
• proof
• proof
• proof : Proof of Theorem \ref{['Ma']}
• proof