The Abelianisation of the real Cremona group

Zimmermann, Susanna. (2015) The Abelianisation of the real Cremona group. Preprints Fachbereich Mathematik, 2015 (34).

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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.