PROFINITE GROUPS WITH A CYCLOTOMIC p-ORIENTATION

Let p be a prime. A continuous representation theta : G -> GL(1) (Z(p)) of a profinite group G is called a cyclotomic p-orientation if for all open subgroups U subset of G and for all k, n >= 1 the natural maps H-k (U, Z(p)(k)/p(N)) -> H-k (U, Z(p)(k)/p) are surjective. Here Z(p)( k ) denotes the Z(p)-module of rank 1 with U-action induced by theta vertical bar(k)(U). By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group G(K) of a field III is, indeed, a cyclotomic p-orientation of G(K). We study profinite groups with a cyclotomic p-orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro-p groups with a cyclotomic p-orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.