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

TL;DR

This paper establishes conditions under which products of certain locally compact groups have fixed price one, resolving a question about $SL(2,Q)$ and employing Cox process methods related to amenable subgroups.

Contribution

It proves fixed price one for products of semisimple Lie groups and p-adic groups, and introduces Cox process techniques to analyze group actions and costs.

Findings

$G_1 imes G_2$ has fixed price one under specified conditions.

All essentially free p.m.p. actions weakly factor onto Cox processes driven by amenable subgroups.

If an amenable subgroup satisfies a double recurrence property, the Cox process driven by it has cost one.

Abstract

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.