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

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a semisimple real Lie group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 2"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> another locally compact second countable unimodular group. We prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1 times upper G 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> × </mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G_1 \times G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has fixed price one if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has higher rank, or if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has rank one and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 2"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic split reductive group of rank at least one. As an application we resolve a question of Gaboriau showing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(2,\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has fixed price one. Inspired by the very recent work of Mikolaj Fraczyk, Sam Mellick, and Amanda Wilkens [ <italic>Poisson-voronoi tessellations and fixed price in higher ranks</italic> , Ann. of Math., to appear], we employ the method developed by the author and Miklós Abért 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.