Some properties of the Cremona group (Q2897169)

The plane Cremona group \(\mathrm{Bir}({\mathbb P}^2)\) is the group made up of birational maps \(\phi:{\mathbb P}^2\dashrightarrow {\mathbb P}^2\) of the complex projective plane \(\mathbb P^2={\mathbb P}({\mathbb C})\). The Noether-Castelnuovo Theorem asserts that \(\mathrm{Bir}({\mathbb P}^2)\) can be generated by quadratic birational maps, i.e., maps defined by homogeneous polynomials of degree 2. Note that the automorphism group \(\mathrm{Aut}({\mathbb C}^2)\) of the complex affine plane \({\mathbb C}^ 2\) may be thought as a subgroup of \(\mathrm{Bir}({\mathbb P}^2)\).NEWLINENEWLINEIn the monograph under review, the author studies some properties of the Cremona group. More precisely, she first gives the Lamy's proof of the amalgamated structure of \(\mathrm{Aut}({\mathbb C}^2)\) (Jung Theorem).NEWLINENEWLINESecond, she discusses a dynamical classification of elements in \(\mathrm{Bir}({\mathbb P}^2)\) according to either the so-called dynamical degree (following J. Diller and C. Favre) or the action of such a map in the so-called Picard-Manin space (following S. Cantat). Then some recent results concerning, for example, Tits alternative (Cantat) or non-simplicity for \(\mathrm{Bir}({\mathbb P}^2)\) (Cantat-Lamy), are described.NEWLINENEWLINENext the author describes the variety of quadratic Cremona maps, where she puts the known classical description in a modern framework.NEWLINENEWLINEThe rest of the monograph is devoted, on one side, to describe other recent results about finite subgroups, Zimmer Conjecture for \(\mathrm{Aut}({\mathbb C}^2)\) and \(\mathrm{Bir}({\mathbb P}^2)\), etc (following M. Kh. Gizatullin, I. Dolgachev, V.A. Iskovskikh, T de Fernex, J. Blanc, S. Cantat and others), on the other side, to describe recent or new results concerning dynamical properties and centralizers of elements in \(\mathrm{Bir}({\mathbb P}^2)\), due to the author and others (E. Bedford, K. Kim, D. Cerveau, C.T.McMullen, etc).