A refined Bloch group and the third homology of SL_2 of a field

TL;DR

This paper investigates the third homology of SL_2 over various fields, showing it is not finitely generated for global fields and providing calculations and bounds for local fields and rings of S-integers.

Contribution

It introduces a refined Bloch group approach to analyze H_3 of SL_2, offering new calculations and bounds for different types of fields.

Findings

H_3 of SL_2 over global fields is not finitely generated

Calculated H_3 of SL_2 over local fields up to 2-torsion

Provided lower bounds for 3-torsion in rings of S-integers

Abstract

We use the properties of the refined Bloch group of a field to prove that H_3 of SL_2 of a global field is never finitely generated, and to calculate - up to some 2-torsion - H_3 of SL_2 of local fields with finite residue field of odd characteristic. We also give lower bounds for the 3-torsion in the H_3 of SL_2 of rings of S-integers.