Journal of Algebra ( IF 0.9 ) Pub Date : 2021-06-08 , DOI: 10.1016/j.jalgebra.2021.06.005 Frauke M. Bleher 1 , Ted Chinburg 2 , Alexander Lubotzky 3
By work of Belyĭ [2], the absolute Galois group of the field of rational numbers can be embedded into , the automorphism group of the free profinite group on two generators. The image of lies inside , the Grothendieck-Teichmüller group. While it is known that every abelian representation of can be extended to , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of . We do this virtually, namely by showing that a rich class of arithmetically defined representations of can be extended to finite index subgroups of . This is achieved, in fact, by extending these representations all the way to finite index subgroups of . 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 .
中文翻译:
关于 Gal(Q‾/Q) , GT^ 和 Aut(F^2) 的表示
通过 Belyĭ [2] 的工作,绝对的伽罗瓦群领域的的有理数可以嵌入到, 自由有限群的自同构群在两台发电机上。的形象躺在里面, Grothendieck-Teichmüller 群。众所周知,每一个阿贝尔表示可以扩展到, Lochak 和 Schneps [13] 提出了构建不可约非阿贝尔表示的挑战. 我们实际上是这样做的,即通过展示丰富的算术定义表示可以扩展到有限索引子群. 事实上,这是通过将这些表示一直扩展到有限索引子群来实现的. 我们通过开发 Grunewald 和 Lubotzky [7] 工作的一个有限版本来做到这一点,它为离散组提供了丰富的表示集合.