Profinite and pro-𝑝 completions of Poincaré duality groups of dimension 3

Abstract

We establish some sufficient conditions for the profinite and pro-<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>completions of an abstract group<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>of type<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F upper P Subscript m"><mml:semantics><mml:mrow><mml:mi>F</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">FP_m</mml:annotation></mml:semantics></mml:math></inline-formula>(resp. of finite cohomological dimension, of finite Euler characteristic) to be of type<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F upper P Subscript m"><mml:semantics><mml:mrow><mml:mi>F</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">FP_m</mml:annotation></mml:semantics></mml:math></inline-formula>over the field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\mathbb {F}_p</mml:annotation></mml:semantics></mml:math></inline-formula>for a fixed natural prime<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>(resp. of finite cohomological<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>-dimension, of finite Euler<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>-characteristic). We apply our methods for orientable Poincaré duality groups<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>of dimension 3 and show that the pro-<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>completion<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</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 a pro-<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>Poincaré duality group of dimension 3 if and only if every subgroup of finite index in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</mml:annotation></mml:semantics></mml:math></inline-formula>has deficiency 0 and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</mml:annotation></mml:semantics></mml:math></inline-formula>is infinite. Furthermore if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</mml:annotation></mml:semantics></mml:math></inline-formula>is infinite but not a Poincaré duality pro-<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>group, then either there is a subgroup of finite index in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</mml:annotation></mml:semantics></mml:math></inline-formula>of arbitrary large deficiency or<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\widehat {G}_p</mml:annotation></mml:semantics></mml:math></inline-formula>is virtually<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\mathbb {Z}_p</mml:annotation></mml:semantics></mml:math></inline-formula>. Finally we show that if every normal subgroup of finite index in<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>has finite abelianization and the profinite completion<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:annotation encoding="application/x-tex">\widehat {G}</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>has an infinite Sylow<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>-subgroup, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:annotation encoding="application/x-tex">\widehat {G}</mml:annotation></mml:semantics></mml:math></inline-formula>is a profinite Poincaré duality group of dimension 3 at the prime<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>.