Maximal deviation of large powers in the Nottingham group

Abstract

The group of formal power series under substitution over Fp, the so-called Nottingham group, is a pro-p group so it is a metric space. The intertwining of these two structures is an important object of study. This paper is concerned with how the power map influences the distance of elements. Suppose that d.f, g/ D d1 while d.f, 1/ D d2 - d1. In this paper we provide a sharp bound for d.f j , gj / in terms of d1, d2 and the exponent j . This bound confirms a conjecture by K. Keating. © 2019 Elsevier B.V., All rights reserved.