2000/02/24 by Ya’acov Peterzil, Anthony L. Pillay, Sergei Starchenko · 1 citation
Mathematics · #Advanced Topology and Set Theory #Rings, Modules, and Algebras #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.1090/s0002-9947-00-02593-9
openalex publication_date 2000/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G=⟨ G, ⋅ ⟩</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">⟨ G,⋅ ⟩</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>