Abelianization of SL₂ over Dedekind domains of arithmetic type

2025/06/14 by Mirzaii, Behrooz, Ramos, Bruno R., Verissimo, Thiago
#11F75 #20J06 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.12638

Abstract

We determine the exact group structure of the abelianization of SL₂(A), where A is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that SL₂(A)^ab is finite, with exponent dividing 12 when char(A)=0, and dividing 6 when char(A)>0. As illustrative cases, we compute SL₂(A)^ab explicitly for instances where A is the ring of integers of a real quadratic field or a cyclotomic extension.

Citations

• The abelianization of the elementary group of rank two
• Linear characters of SL₂ over Dedekind domains
• The homological torsion of PSL₂ of the imaginary quadratic integers

Related

• Linear characters of SL₂ over Dedekind domains
• The abelianization of SL₂(ℤ[(1)/(m)])
• Relative Galois module structure of rings of integers of absolutely Abelian number fields
• On finite and elementary generation of SL₂(R)
• The abelianization of the elementary group of rank two