The plane Cremona Group $Cr_2(\mathbb{C})$ is the group of birational automorphisms of $\mathbb{P^2_{\mathbb{C}}}$. There are a lot of articles about this group (over $\mathbb{C}$). But I can't find anything about $Cr_2(\mathbb{R})$. What is known about this group? Why it's so difficult to study this group?

In this article: http://arxiv.org/abs/1306.6063 you can find some descriptions of the generators of the Cremona group over the reals, and get some other results. See also the references inside. 

