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On the groups of birational transformations of surfaces. (Sur les groupes de transformations birationnelles des surfaces.) (French. English summary) Zbl 1233.14011
Let \(S\) be a complex compact Kähler surface and denote by \(\text{Bir}(S)\) the group of birational transformations of \(S\); when \(S\) is not projective it admits a unique minimal model \(S_0\) and \(\text{Bir}(S)=\operatorname{Aut}(S_0)\) is the automorphism group of \(S_0\). In the paper under review the author investigates properties of subgroups in \(\text{Bir}(S)\). He obtains (essentially) three main results.
First he proves that if \(\Gamma\subset \text{Bir}(S)\) is a countable subgroup and every action of \(\Gamma\) by affine isometries on a Hilbert space admits a fixed point, i.e., if \(\Gamma\) satisfies the so-called Kazhdan (T) property, then either \(\Gamma\) is finite or there exists a birational map \(S \dashrightarrow \mathbb P^2\) which conjugates \(\Gamma\) to a subgroup of automorphisms of the complex projective plane (Theorem A in the paper). As a corollary, he obtains an analogue of the Zimmer conjecture on lattices in real Lie groups acting on compact manifolds, in the case where actions are birational and manifolds are compact Kähler surfaces.
Secondly, for an element \(f\in\text{Bir}(S)\) whose first dynamical degree (e.g. when \(S\) is the complex projective plane it is the limit of the sequence \(\sqrt[n]{\deg(f^n)}\), where \(\deg\) means algebraic degree) is greater than 1, he proves that if \(G\subset\text{Bir}(S)\) is a subgroup whose elements commute with \(f\), then it is an extension of a torsion group by a multiplicative subgroup of positive real numbers; moreover, this last subgroup is cyclic when \(G\) is finitely generated. In particular he deduces that if \(g\in\text{Bir}(S)\) commutes with \(f\) then there exist integers \(m>0\) and \(n\) such that \(g^m=f^n\) (Theorem B in the paper).
Finally, the author proves the so-called Tits alternative for \(\text{Bir}(S)\), that is, he shows that every finitely generated subgroup of \(\text{Bir}(S)\) contains either a non abelian free subgroup or a soluble subgroup of finite index (Theorem C in the paper). As a corollary he deduces, among other things, that every finitely generated torsion subgroup of \(\text{Bir}(S)\) is finite.
The reader may also consult [Astérisque 332, 11–43, Exp. No. 998 (2010; Zbl 1210.14015)] by C. Favre.

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
32M05 Complex Lie groups, group actions on complex spaces
20E36 Automorphisms of infinite groups
20F28 Automorphism groups of groups
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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