Galois groups and fundamental groups

If X is a geometrically connected algebraic variety over a field k，X - D4- D4 =X【cross product】k - D4- D4, k - D4 being an algebraic closure of k，then the absolute Galois group G - D2k - D2=Gal(k - D4- D4/k) of k acts outerly on the algebraicfundamental group π D21 - D2(X - D4- D4) of X - D4- D4 in a natural manner.The head investigatorIhara continued his study on the arithmetic aspects of this action in the most basic case where k=Q（the rational number filed） and X=P y D11 y D1-{0,1,∞}(the projective line minus three points). Inthis case, π D21 - D2 (X - D4- D4) is a free profinite group (F - D4^ D4) - D22 - D2 of rank 2，on which G - 2q - D2 acts faithfully. G - D2Q - D2 acts faithfullyregarded as a subgroup of the automorphism group Aut(F - D4^ D4) D22 - D2，is known to be contained in a subgroup GT of Aut(F - D4^ D4) d - 22 D2 （the Grothendieck-Teichmuller）group). It is not known whether G D2Q D2≠GT，but there are some properties known to be satisfied by elements of gi D2Q D2 but unkno…wn and doubtful whether they are satisfied by "any" element of GT.Ihara completed his More and doubtful whether they are satisfied by "any" element of GT.Ihara completed hisstudy of the GT-action on the maximal meta-abelian quotient of (F - D4^ D4) d - 22 D2 [1]. Itconcerns with the theory of adelic beta functions and their г-decompositions，continuing previous works of G. Anderson and Ihara himself. This contains a certain arithmetic必要条件for an element σ∈GT to belong to G d - 2q D2 in terms of the （hyper adelic）gamma function г D2σ D2 and certain quasi 1-cocycles. In [2]，more basic arithmeticconditions in terms of quasi 1-cocycles and geometric conditions反射of the circumstances that cyclic covers of X - D4- D4可以作为开放使用subspaces of X - D4) are discussed，and logical dependencies among these conditions are clarified.In[3]，Ihara studied the gi D2Q D2Q D2 action ψ D1 on the maximal prop-p quotientF - D3(p - D3), d - 22 d - D2 of (F - D4^ - D4) d - 22 d - D2 (p；an arbitrary fixed prime)in connection with (i) abelian extensions over the cyclotomic filed Q(μ D2p D2∞),(ii) the stablederivation algebra. Among them, (i) concerns with the kernel of ψ D1pasking which abelian extension of Q(μ D2p D2∞)is contained in the field corresponding to thiskernel, while (ii)is a sort of the "graded Lie algebra version over Z" of GT，and is a basic object in studying the image of ψ。the main innovation of [3] is theclarification of the connection between (i) (ii)，and some numerical result which shows quite an exciting phenomenon related to (i) (ii) when p is anirregular prime. Less

If X is a geometrically connected algebraic variety over a field k, and XィイD4-ィエD4 =X 【cross product】 kィイD4-ィエD4, kィイD4-ィエD4 being an algebraic closure of k, then the absolute Galois group GィイD2kィエD2=Gal(kィイD4-ィエD4/k) of k acts outerly on the algebraic fundamental group πィイD21ィエD2(XィイD4-ィエD4) of XィイD4-ィエD4 in a natural manner.The head investigator Ihara continued his study on the arithmetic aspects of this action in the most basic case where k=Q (the rational number filed) and X=PィイD11ィエD1-{0,1,∞} (the projective line minus three points). In this case, πィイD21ィエD2 (XィイD4-ィエD4) is a free profinite group (FィイD4^ィエD4)ィイD22ィエD2 of rank 2, on which GィイD2QィエD2 acts faithfully. GィイD2QィエD2, regarded as a subgroup of the automorphism group Aut(FィイD4^ィエD4)ィイD22ィエD2, is known to be contained in a subgroup GT of Aut(FィイD4^ィエD4)ィイD22ィエD2 (the Grothendieck-Teichmuller group). It is not known whether GィイD2QィエD2≠GT, but there are some properties known to be satisfied by elements of GィイD2QィエD2 but unkno … More wn and doubtful whether they are satisfied by "any" element of GT.Ihara completed his study of the GT-action on the maximal meta-abelian quotient of (FィイD4^ィエD4)ィイD22ィエD2 [1]. It concerns with the theory of adelic beta functions and their г-decompositions, continuing previous works of G. Anderson and Ihara himself. This contains a certain arithmetic necessary condition for an element σ ∈ GT to belong to GィイD2QィエD2 in terms of the (hyper adelic) gamma function гィイD2σィエD2 and certain quasi 1-cocycles. In [2], more basic arithmeticconditions (in terms of quasi 1-cocycles) and geometric conditions (a reflection of the circumstances that cyclic covers of XィイD4-ィエD4 can also be embedded as open subspaces of XィイD4-ィエD4) are discussed, and logical dependencies among these conditions are clarified.In[3], Ihara studied the GィイD2QィエD2 action ψィイD1(pィエD1) on the maximal prop-p quotient FィイD3(pィエD3),ィイD22ィエD2 of (FィイD4^ィエD4)ィイD22ィエD2 (p ; an arbitrary fixed prime), in connection with (i) abelian extensions over the cyclotomic filed Q(μィイD2pィエD2∞),(ii) the stable derivation algebra. Among them, (i) concerns with the kernel of ψィイD1pィエD1; asking which abelian extension of Q(μィイD2pィエD2∞) is contained in the field corresponding to this kernel, while (ii)is a sort of the "graded Lie algebra version over Z" of GT, and is a basic object in studying the image of ψィイD1(pィエD1). The main innovation of [3] is the clarification of the connection between (i) (ii), and some numerical result which shows quite an exciting phenomenon related to (i) (ii) when p is an irregular prime. Less