K₁-congruences for three-dimensional Lie groups

Abstract

We completely describe K₁ (Zₚ [[G∞]]) and its localisations by using an infinite family of p-adic congruences, where G∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G∞)=2 , and of the first named author and Lloyd Peters when G∞≅Z×p⋉Z𝒹ₚ with a scalar action of Z×ₚ . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory.