On the vanishing of $SK_1$ of certain Iwasawa algebras

TL;DR

This paper proves the vanishing of the $SK_1$ group for specific Iwasawa algebras associated with certain pro-$p$ groups, advancing understanding in algebraic K-theory and Iwasawa theory.

Contribution

It establishes the vanishing of $SK_1$ for Iwasawa algebras of particular pro-$p$ groups, a result not previously known for these cases.

Findings

$SK_1( ext{Z}_p ext{[[}G ext{]]})$ vanishes for $G= ext{Z}_p^n times ext{Z}_p$

$SK_1( ext{Z}_p ext{[[}G ext{]]})$ vanishes for Heisenberg pro-$p$ groups

Advances understanding of algebraic K-theory in Iwasawa algebras

Abstract

We prove the vanishing of $SK_1(\Zp\ps{G})$ when $G$ is either $\Zp^n\rtimes\Zp$ or a Heisenberg pro-$p$ group.