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SUMMARY:Frauke Bleher (University of Iowa)
DTSTART:20200926T180000Z
DTEND:20200926T184000Z
DTSTAMP:20260423T024452Z
UID:SheRepTh/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SheRepTh/11/
 ">Automorphism groups of pro-free groups and their representations</a>\nby
  Frauke Bleher (University of Iowa) as part of Sherbrooke Meeting on Repre
 sentation Theory of Algebras\, Corona Edition (fully online)\n\n\nAbstract
 \nThis is joint work with Ted Chinburg and Alex Lubotzky. Let $d\\ge 2$\, 
 let $F_d$ be a free discrete group of rank $d$\, and let $\\hat{F}_d$ be i
 ts profinite completion. Grunewald and Lubotzky developed a method to cons
 truct\, under some technical conditions\, representations of finite index 
 subgroups of $\\mathrm{Aut}(F_d)$ that have as images certain large arithm
 etic groups. In this talk\, I will first show how their method leads to a 
 stronger result for $\\mathrm{Aut}(\\hat{F}_d)$. I will then discuss an ap
 plication of this result to Galois theory. This uses a result by Belyi who
  showed that there is a natural embedding of the absolute Galois group $G_
 {\\mathbb{Q}}$ of $\\overline{\\mathbb{Q}}$ over $\\mathbb{Q}$ into $\\mat
 hrm{Aut}(\\hat{F}_2)$.  In particular\, I will show how the natural action
  of certain subgroups of $G_{\\mathbb{Q}}$ on the Tate modules of generali
 zed Jacobians of covers of $\\mathbb{P}^1$ over $\\overline{\\mathbb{Q}}$ 
 that are unramified outside $\\{0\,1\,\\infty\\}$ can be extended\, up to 
 a finite index subgroup\, to an action of a finite index subgroup of $\\ma
 thrm{Aut}(\\hat{F}_2)$. If time permits\, I will give a criterion for this
  action to define\, up to a finite index subgroup\, a compatible action on
  the Tate modules of the usual Jacobians of the covers.\n
LOCATION:https://researchseminars.org/talk/SheRepTh/11/
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