the boundary of a growing ball, a ball starts to grow at this point until it has radius $2$ and so on. Let $L_t^{\lambda}(x)$ denote the set which has been passed by the boundary of some ball at time $t$. If $C^{\lambda}(x)$ is bounded, then $L_t^{\lambda}(x)$ stops growing at some random time $T$. In this case $C^{\lambda}[L_T^{\lambda}(x)]=C^{\lambda}(x)$ and $L_T^{\lambda}(x)$ is the $1$-neighbourhood of $C^{\lambda}(x)$. (If the first ball does not hit any Poisson point, then $C^{\lambda}(x)$ is the empty set). If $C^{\lambda}(x)$ is unbounded, then $L_t^{\lambda}(x)$ never stops growing. We will refer to this procedure to as “growing the component containing $x$”.

In what follows we will make use of the following lemma, which may be considered intuitively clear. The proof is inspired by the proof of the corresponding lemma for the discrete situation which is Lemma 1.1 of [@schonmann].

\[ballemma\] Consider the Poisson Boolean model on a homogeneous space $M$. Let $R>0$ and let $\lambda>\lambda_c$. Any unbounded component of $C^{\lambda}$ contains balls of radius $R$.

For the proof we need to introduce some further notation. For a connected set $A$ containing $x$ we let $C^{\lambda}(x,A)$ be all points in $A$ which can be connected to $x$ by some curve 