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Normal subgroups in the Cremona group. (English) Zbl 1278.14017

The Main Theorem of the paper asserts that over an algebraically closed field \(k\), the so-called Cremona group \(\mathrm{Bir}({\mathbb P}_k^2)\), made up of birational maps \({\mathbb P}_k^2\dashrightarrow {\mathbb P}_k^2\) of the projective plane \({\mathbb P}_k^2\), is not simple.
The strategy consists in doing \(\mathrm{Bir}({\mathbb P}_k^2)\) to act faithfully by isometries on an infinite-dimensional hyperbolic space and introducing a certain type of birational map, said to be tight, which acts as a hyperbolic isometry and whose large enough powers are shown to generate a proper normal subgroup.
Then the authors prove, on one side, that if \(g\in \mathrm{Bir}({\mathbb P}_k^2)\) is a general element among those which transform (birationally) a pencil of lines into a pencil of lines, then \(g\) is tight. On the other side, they construct automorphisms of Coble and Kummer (rational) surfaces which correspond to tight elements in \(\mathrm{Bir}({\mathbb P}_k^2)\).
The paper under review also contains an appendix by Yves de Cornulier who proves that if \(\mathrm{Bir}({\mathbb P}_k^2)\) acts by isometries on a complete real tree, then all its elements admit a fixed point; as a consequence he obtains that the Cremona group cannot be written as a non-trivial amalgamated product.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
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