The Abelianisation of the real Cremona group

We present the Abelianisation of the birational transformations of $\mathbb{P}_{\mathbb{R}}^2$. Its kernel is equal to the normal subgroup generated by $PGL_3(\mathbb{R})$, and contains all elements of degree $\le 4$. The description of the quotient yields the existence of normal subgroups of index $2^n$ for any n and that the normal subgroup generated by any countable set of elements is a proper subgroup.