Let $d>0$ be a square free positive integer and let $\mathcal{O}_d$ be the ring of integers in $\mathbb{Q}[\sqrt{d}]$. What is the abelianization of the Hilbert modular group $\text{SL}_2(\mathcal{O}_d)$? If this is too hard, is at least the rank of the abelianization known?

I'd also be interested in knowing this for finite-index subgroups of $\text{SL}_2(\mathcal{O}_d)$.

These groups are lattices in $\text{SL}_2(\mathbb{R}) \times \text{SL}_2(\mathbb{R})$. I believe that this implies that they don't have property (T), so there isn't a cheap way of seeing that the rank of the abelianization is $0$. But they do have *some* higher-rank behavior; for instance, Serre proved that they do have the congruence subgroup property.