Internal Indecomposability of Various Profinite Groups in Anabelian Geometry

Abstract

It is well-known that various profinite groups appearing in anabelian geometry satisfy distinctive group-theoretic properties such as the slimness [i.e., the property that every open subgroup is center-free] and the strong indecomposability [i.e., the property that every open subgroup has no nontrivial product decomposition]. In the present paper, we consider another group-theoretic property on profinite groups, which we shall refer to as strong internal indecomposability --this is a stronger property than both the slimness and the strong indecomposability-- and prove that various profinite groups appearing in anabelian geometry [e.g., the étale fundamental groups of hyperbolic curves over number fields, p-adic local fields, or finite fields; the absolute Galois groups of Henselian discrete valuation fields with positive characteristic residue fields or Hilbertian fields] satisfy this property. Moreover, by applying the pro-prime-to-p version of the Grothendieck Conjecture for hyperbolic curves over finite fields of characteristic p [established by Saidi and Tamagawa], together with some considerations on almost surface groups, we also prove that the Grothendieck-Teichmüller group satisfies the strong indecomposability . This gives an affirmative answer to an open problem posed in a first author's previous work.