Condensation and left-orderable groups

We discuss condensed left-orderings and develop new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. These techniques apply to the solvable Baumslag Solitar groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper S left-parenthesis 1 comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">BS(1,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to Thompson’s group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .