When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

Question

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is known in a more general setting. What is the most general setting for which the answer is known? In particular does it hold for all locally compact second countable hausdorff topological groups?

Answer 1

What appears to be the key to the D&V-J proof is that there is a non-empty open set $U\subset\Bbb R^d$ sand an infinite countable collection of points $\{x_n\}$ such that the translates $\{x_n+U: n\in\Bbb N\}$ are disjoint. If your LCCB group has this property, then the proof will still work.