Fesenko group

In mathematics, Fesenko groups are certain subgroups of the wild automorphism groups of local fields of positive characteristic (i.e. the Nottingham group), studied by Ivan Fesenko (Fesenko (1999)).

The Fesenko group F(F_p) is a closed subgroup of the Nottingham group N(F_p) consisting of formal power series t + a₂t^1+2p+a₃t^1+3p+... with coefficients in F_p. The group multiplication is induced from that of the Nottingham group and is given by substitution.

The group multiplication is not abelian. This group is torsion free (Fesenko (1999)), unlike the Nottingham group. This group is a finitely generated pro-p-group and a hereditarily just infinite group (Fesenko (1999)). Thus, it is another representative of the 4th class of hereditarily just infinite groups, together with the Nottingham group and the Grigorchuk group, according to the conjectural classification of his group by Charles Leedham-Green. The Fesenko group is of finite width (Griffin (2005)). It can be realized as the Galois group of an arithmetically profinite extension of local fields (Fesenko (1999)), while it is stille unknown whether the Nottingham groups shares the same property.

References

• Fesenko, Ivan (1999), "On just infinite pro-p-groups and arithmetically profinite extensions of local fields", Journal für die reine und angewandte Mathematik, 517: 61–80, arXiv:math/9802092, doi:10.1515/crll.1999.098, ISSN 0075-4102, MR 1728547
• Griffin, Cornelius (2005), "The Fesenko groups have finite width", The Quarterly Journal of Mathematics, 56 (3): 337–344, arXiv:math/0310038, doi:10.1093/qmath/hah033, ISSN 0033-5606, MR 2161247
• du Sautoy, Marcus; Fesenko, Ivan (2000), "Where the wild things are: ramification groups and the Nottingham group", in du Sautoy, Marcus; Segal, Dan; Shalev, Aner (eds.), New horizons in pro-p groups, Progr. Math., 184, Boston, MA: Birkhäuser Boston, pp. 287–328, ISBN 978-0-8176-4171-9, MR 1765121