Invariable generation of certain groups of piecewise linear homeomorphisms of the interval

<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the group of all the orientation preserving piecewise linear homeomorphisms of the interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="a greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P Superscript a"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">P^a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of all the elements with slopes in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="a Superscript double-struck upper Z"> <mml:semantics> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">a^\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P Superscript double-struck upper Q"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">P^\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of all the elements with slopes and breaks in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P Superscript a"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">P^a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper P Superscript double-struck upper Q"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">P^\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as Thompson 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>, are invariably generated. </p>