On representations of Gal(Q‾/Q), GTˆ and Aut(Fˆ2)
| Autor: | Alexander Lubotzky, Ted Chinburg, Frauke M. Bleher |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Rational number
Algebra and Number Theory Profinite group Group (mathematics) Discrete group Image (category theory) 010102 general mathematics Field (mathematics) Absolute Galois group 01 natural sciences Combinatorics 0103 physical sciences 010307 mathematical physics 0101 mathematics Abelian group Mathematics |
| Zdroj: | Journal of Algebra. 607:134-159 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2021.06.005 |
| Popis: | By work of Belyĭ [2] , the absolute Galois group G Q = Gal ( Q ‾ / Q ) of the field Q of rational numbers can be embedded into A = Aut ( F ˆ 2 ) , the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ , the Grothendieck-Teichmuller group. While it is known that every abelian representation of G Q can be extended to G T ˆ , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ . We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ . This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut ( F ˆ 2 ) . We do this by developing a profinite version of the work of Grunewald and Lubotzky [7] , which provided a rich collection of representations for the discrete group Aut ( F d ) . |
| Databáze: | OpenAIRE |
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