to be a (Riemannian) homogeneous space if for each $x,y\in M$ there is an isometry that takes $x$ to $y$. Throughout this paper we assume that $M$ is an unbounded homogeneous space, with metric $d_M$ and volume measure $\mu_M$. When it is clear which space we are working with we will write $d=d_M$ and $\mu=\mu_M$. We let $0$ denote the origin of the space.

For one of the main results below it is possible to give a shorter proof under the additional assumption that $M$ is a symmetric space. A connected Riemannian manifold $M$ is said to be a (Riemannian) symmetric space if for each point $p\in M$ there is an isometry $I_p$ such that $I_p(p)=p$ and $I_p$ reverses geodesics through $p$. The most important symmetric spaces where it makes sense to study continuum percolation are arguably $n$-dimensional Euclidean space ${\mathbb R}^n$ and $n$-dimensional hyperbolic space $\hypn$. Also products of symmetric spaces are symmetric spaces, for example $\hyp\times {\mathbb R}$. Any symmetric space is homogeneous. For an example of a noncompact space which is homogeneous but not symmetric, one may consider certain Damek-Ricci spaces, see [@berndt]. Next we introduce the Poisson Boolean model of continuum percolation.

Let $S(x,r):=\{y\in M\,:\,d_M(x,y)\le r\}$ 