A splitting for K_1 of completed group rings

TL;DR

This paper constructs a splitting of the K_1 map for completed group rings related to pro-p Lie groups and proves the vanishing of SK_1 for specific unipotent groups, advancing algebraic K-theory and Iwasawa theory.

Contribution

It introduces a new splitting construction for K_1 groups in the context of completed group rings and establishes SK_1 vanishing results for unipotent groups.

Findings

Constructed a splitting of the K_1 map from mod p reduction of Iwasawa algebra.

Proved SK_1 vanishes for certain unipotent groups.

Linked the results to Coleman power series and fields of norms.

Abstract

Motivated by the theory of Coleman power series (reinterpreted via fields of norms by Fontaine) we construct a splitting of the natural map of K_1 groups arising from the mod p reduction map of the Iwasawa algebra of a pro-p Lie group. We also show the vanishing of SK_1 for certain unipotent groups.