Third homology of $\mathrm{SL}_{2}$ over Number fields: The norm-Euclidean quadratic imaginary case

TL;DR

This paper extends the understanding of the third homology of SL2 over number fields, specifically for norm-Euclidean quadratic imaginary fields, by developing properties of the refined scissors congruence group.

Contribution

It introduces new properties of the refined scissors congruence group to compute the third homology of SL2 over certain imaginary quadratic fields.

Findings

Extended the structure of third homology to norm-Euclidean quadratic imaginary fields.

Connected the homology to refined scissors congruence groups.

Provided new algebraic tools for homology calculations.

Abstract

In the article The third homology of $SL_{2}(\mathbb{Q})$, Hutchinson determined the structure of $H_{3}\left(\mathrm{SL}_{2}(\mathbb{Q}),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ by expressing it in terms of $K_{3}^{\mathrm{ind}}(\mathbb{Q})\cong \mathbb{Z}/24$ and the scissor congruence group of the residue field $\mathbb{F}_{p}$ with $p$ a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.