ImpactU - Detalle del Producto

On the procongruence completion of the Teichmüller modular group

Acceso Abierto

Idioma: Inglés

Publicado: 12/05/2014

APC (est): No disponible

Abstract:

For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 g minus 2 plus n greater-than 0"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2g-2+n>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Teichmüller modular group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Gamma _{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a compact Riemann surface of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points removed, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the group of homotopy classes of diffeomorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which preserve the orientation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a given order of its punctures. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Pi _{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the fundamental group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with a given base point, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Pi With caret Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\hat {\Pi }_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> its profinite completion. There is then a natural faithful representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript g comma n Baseline right-arrow with hook normal upper O normal u normal t left-parenthesis ModifyingAbove normal upper Pi With caret Subscript g comma n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">↪</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">O</mml:mi> <mml:mi mathvariant="normal">u</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma _{g,n}\hookrightarrow \mathrm {Out}(\hat {\Pi }_{g,n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>The procongruence Teichmüller group</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Gamma With ˇ Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">ˇ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\check {\Gamma }_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined to be the closure of the Teichmüller group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Gamma _{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> inside the profinite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper O normal u normal t left-parenthesis ModifyingAbove normal upper Pi With caret Subscript g comma n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">O</mml:mi> <mml:mi mathvariant="normal">u</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Out}(\hat {\Pi }_{g,n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we begin a systematic study of the procongruence completion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Gamma With ˇ Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">ˇ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\check {\Gamma }_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The set of <italic>profinite Dehn twists</italic> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Gamma With ˇ Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">ˇ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\check {\Gamma }_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the closure, inside this group, of the set of Dehn twists of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Gamma _{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Gamma With ˇ Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">ˇ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\check {\Gamma }_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the subsequent description of their centralizers (Sections 5 and 6). This is the basis for the Grothendieck-Teichmüller Lego with procongruence Teichmüller groups as building blocks. As an application, in Section 7, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful.

Tópico:

Algebraic Geometry and Number Theory

Citaciones:

Citaciones por año:

Altmétricas:

Información de la Fuente:

FuenteTransactions of the American Mathematical Society	Cuartil año de publicaciónNo disponible	Volumen366
Issue10	Páginas5185 - 5221	pISSNNo disponible
ISSN1088-6850	Perfil OpenAlexhttps://openalex.org/S60030702

Enlaces e Identificadores:

Openalex URL	https://openalex.org/W2963303873	Doi URL	https://doi.org/10.1090/s0002-9947-2014-05971-0	Pdf URL	https://www.ams.org/tran/2014-366-10/S0002-9947-2014-05971-0/S0002-9947-2014-05971-0.pdf
Open_access URL	https://doi.org/10.1090/s0002-9947-2014-05971-0	Scholar citations URL	https://scholar.google.com/scholar?cites=15874622264403491659&as_sdt=2005&sciodt=0,5&hl=en	Scholar URL	https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=info%3AS_tGldf8TdwJ%3Ascholar.google.com&btnG=

Artículo de revista