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SUMMARY:Thomas Weigel (Università di Milano Bicocca)
DTSTART:20210520T150000Z
DTEND:20210520T160000Z
DTSTAMP:20260423T053015Z
UID:GOThIC/32
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GOThIC/32/">
 Maximal pro $p$-quotients of absolute Galois groups</a>\nby Thomas Weigel 
 (Università di Milano Bicocca) as part of GOThIC - Ischia Online Group Th
 eory Conference\n\n\nAbstract\n(Joint work with Claudio Quadrelli.)\n\nIt 
 is well-known that the absolute Galois group $G_K = \\operatorname{Gal}(\\
 bar K^{\\text{sep}}/K)$ of a field $K$\nis a profinite group. However\, on
 ly in very restrictive circumstances it is possible to\nanalyze the struct
 ure of $G_K$ completely. A first approximation - which is untertaken\nfreq
 uently - is to investigate the maximal pro-$p$ quotient $G_K(p) = G_K/O^{p
 }\n(G_K)$ for a prime $p$. Here $O_p(\\_)$ is the closed subgroup being ge
 nerated by all Sylow\npro-$\\ell$ subgroups for $\\ell \\ne p$. The absolu
 te Galois group $G_K$ comes equipped with a\ncontinuous group homomorphism
 \n$$\n\\theta_{K\,p} : G_K \\to \\mathbb{Z}_p^{x}\n\,$$\nthe $p$-cyclotomi
 c character\, where $\\mathbb{Z}_p^{x}$ denotes group of the invertible el
 ements in\nthe ring of $p$-adic integers $\\mathbb{Z}_p$. In case that $K$
  contains a primitive $p$-th root of unity\,\nthe homomorphism $\\theta_{K
 \,p}$ is induced from a group homomorphism\n$$\n\\hat\\theta_{K\,p} : G_K(
 p) \\to \\mathbb{Z}_p^{x}\n.$$\n\nA pro-$p$ group $G$ together with a cont
 inuous group homomorphism $\\theta : G → \\mathbb{Z}_p^{x}$\nis\nalso ca
 lled an oriented pro-$p$ group. Although the structure of $G_K(p)$ is in g
 eneral\nmuch easier to analyze than $G_K$ there are still many open questi
 ons concerning\nthe oriented pro-$p$ groups $(G_K(p)\, \\hat\\theta_{K\,p}
 )$. E.g.\, around 25 years ago it was conjectured by I. Efrat\, that in ca
 se that $G_K(p)$ is a finitely generated pro-$p$ group\,\nthen $(G_K(p)\, 
 \\hat\\theta_{K\,p})$ must be of elementary type. Here one defines the cla
 ss of oriented pro-$p$ groups of elementary type as the smallest class of 
 oriented pro-$p$ groups\nwhich is closed under free products\, and fibre p
 roducts with $\\theta$-abelian oriented pro-$p$ groups which contains $(F\
 , \\alpha)$ for all finitely generated free pro-$p$ groups $F$ and any $\\
 alpha : F \\to \\mathbb{Z}_p^{x}$\, as well as $(D\, \\eth)$ for all Demus
 h’kin pro-$p$ groups $D$\, where $\\eth: D \\to \\mathbb{Z}_p^{x}$ is th
 e $p$-orientation induced by the dualizing module of D. In the talk I will
  discuss recent developments in Field theory\, which transformed I. Efrat
 ’s elementary\ntype conjecture into a purely group theoretic question. R
 ecently\, this question\nhas been investigated successfully for certain cl
 asses of oriented pro-$p$ groups: 1)\nRight-angled Artin pro-$p$ groups wi
 th trivial orientation (I. Snopce\, P. Zalesskii)\,\n2) Generalized right-
 angled Artin pro-$p$ groups (S. Blumer\, C. Quadrelli\, T.W.).\n
LOCATION:https://researchseminars.org/talk/GOThIC/32/
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