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• W3048702928 abstract "We develop a correspondence between Borel equivalence relations induced by closed subgroups of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript normal infinity> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=approximately-equals Subscript omega plus 1 comma 0 Superscript asterisk Baseline> <mml:semantics> <mml:msubsup> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>cong ^ast _{omega +1,0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly below <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=approximately-equals Subscript omega plus 1 comma greater-than omega Superscript asterisk Baseline> <mml:semantics> <mml:msubsup> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>cong ^ast _{omega +1,>omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in Borel reducibility. By results of Hjorth-Kechris-Louveau, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=approximately-equals Subscript omega plus 1 comma greater-than omega Superscript asterisk Baseline> <mml:semantics> <mml:msubsup> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>cong ^ast _{omega +1,>omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provides invariants for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript omega plus 1 Superscript 0> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Sigma ^0_{omega +1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equivalence relations induced by actions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript normal infinity> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, while <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=approximately-equals Subscript omega plus 1 comma 0 Superscript asterisk Baseline> <mml:semantics> <mml:msubsup> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>cong ^ast _{omega +1,0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provides invariants for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript omega plus 1 Superscript 0> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Sigma ^0_{omega +1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equivalence relations induced by actions of <italic>abelian</italic> closed subgroups of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript normal infinity> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation <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>, Borel bireducible with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=equals Superscript plus plus Baseline> <mml:semantics> <mml:msup> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>=^{++}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, so that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F up harpoon with barb right upper C> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>↾<!-- ↾ --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Frestriction C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not Borel reducible to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=equals Superscript plus Baseline> <mml:semantics> <mml:msup> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>=^{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any non-meager set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal (2013). For these proofs we analyze the symmetric models <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Subscript n> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>M_n</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=n greater-than omega> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>n>omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, developed by Monro (1973), and extend the construction past <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega> <mml:semantics> <mml:mi>ω<!-- ω --></mml:mi> <mml:annotation encoding=application/x-tex>omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, through all countable ordinals. This answers a question of Karagila (2019)." @default.
• W3048702928 created "2020-08-18" @default.
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• W3048702928 date "2020-11-03" @default.
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• W3048702928 title "Borel reducibility and symmetric models" @default.
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