Just infiniteness and other properties of the generalized Nottingham group

The generalized Nottingham group was introduced over finite fields in 95 by Shalev, aiming to find new examples of just infinite profinite groups. In this thesis a definition over arbitrary rings is shown and some of its properties are proved. In particular it is proved that the generalized Nottingham group over a finite field of odd characteristic is hereditarily just infinite. Moreover some families of subgroups with remarkable properties are introduced, among which the Cartan type groups (already informally defined by Shalev), the pseudo-algebraic groups and the index-subgroups. The latter family is an extension of the family of the same name introduced by Barnea and Klopsch and allows us to prove an analogous of a result proved by them. Finally it is proved that any k-tuple of randomly chosen element of the generalized Nottingham group generate an abstract free group of rank k with probability 1, generalizing a result by Szegedy.