Quelques propriétés des sous-groupes de Gal(k((t))/k)

Abstract

This article is devoted to the study of some structural properties of subgroups of Gal(k((t))/k), group of k-automorphisms of k((t)), the Laurent series field on a commutative field k. We start showing that the center of Gal(k((t))/k) is trivial in every characteristic. Then, assuming chark=0, we show that the center of any non-abelian subgroup of Gal(k((t))/k) is always a cyclic group, property that is called (Zc) for any group G. This last result is given by closely investigating the centralizer of any element of Gal(k((t))/k). This study is based on the introduction of raising an automorphism to the power a (for a∈k) which is defined on principal automophisms subgroup (consisting of σ∈Gal(k((t))/k) satisfying v(σ(t)−t)≥2). In particular, alongside this study we show that this last subgroup is a CA-group.We then turn our interest for groups having (Zc) property. We show that the free product with amalgamation of two groups having (Zc) property over a third group has continue having (Zc) property. This results leads us to wonder that given two torsion elements of Gal(k((t))/k) whether or not the subgroup generated by these two elements is a free product with amalgamation. We also provide a graphical interpretation of this problem in connection with Serre–Bass theory on trees and we give two different situations on such subgroups, one where the answer is positive and another situation with a negative answer.