Gaboriau's criterion and fixed price one for locally compact groups

Let $G_1$ be a semisimple real Lie group and $G_2$ another locally compact second countable unimodular group. We prove that $G_1 \times G_2$ has fixed price one if $G_1$ has higher rank, or if $G_1$ has rank one and $G_2$ is a $p$-adic split reductive group of rank at least one. As an application we resolve a question of Gaboriau showing $SL(2,\mathbb{Q})$ has fixed price one. Inspired by the very recent work arXiv:2307.01194v1 [math.GT], we employ the method developed by the author and Mikl\'os Ab\'ert to show that all essentially free probability measure preserving actions of groups weakly factor onto the Cox process driven by their amenable subgroups. We then show that if an amenable subgroup can be found satisfying a double recurrence property then the Cox process driven by it has cost one.