The Cremona Group in 2 Dimensions
In two dimensions, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with PGL(3, k), though there was some controversy about whether their proofs were correct, and Gizatullin (1983) gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it.
- Cantat & Lamy (2010) showed that the Cremona group is not simple as an abstract group;
- Blanc showed that it has no normal subgroups other than the trivial group and itself that are also closed in a natural topology.
- For the finite subgroups of the Cremona group see Dolgachev & Iskovskikh (2009).
Read more about this topic: Cremona Group
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