On solvable subgroups of the Cremona group

Abstract

The Cremona group Bir(P 2 C ) is the group of birational self-maps of P 2 C . Using the action of Bir(P 2 C ) on the Picard-Manin space of P 2 C we characterize its solvable subgroups. If G ⊂ Bir(P 2 C ) is solvable, non virtually abelian, and infinite, then up to finite index: either any element of G is of finite order or conjugate to an automorphism of P 2 C , or G preserves a unique fibration that is rational or elliptic, or G is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.We also give some corollaries.

Tits-type Alternative for Groups Acting on Toric Affine Varieties

“…The next theorem is a reformulation of [Des21, Theorem 8.49], which in turn integrates the results of [Des15] and [Ure21, Theorem 7.1] based in particular on [Can11, Proposition 6.14 and Theorem 7.7].…”

Subgroups of elliptic elements of the Cremona group

“…Solvable subgroups. In [Dés15], Déserti gives a description of solvable subgroups of Cr 2 (C). Theorem 1.3 and 1.5 refine her results.…”

“…Remark 7.2. It seems that in the proof of the main theorem in [Dés15] it has not been considered that a priori there could be loxodromic elements with different axes but a common fixed point on ∂ H ∞ (in this case the ping-pong Lemma can not be applied). However, Lemma 7.3 fills this gap by showing that such loxodromic elements do not exist.…”

Birational Kleinian groups

“…The bulk of the classification is done by Cantat [Can11] and it is finished by Urech [Ure21]. Besides [Can11] and [Ure21], the works of Weil [Wei55], Gizatullin [Giz80], Blanc-Cantat [BC16] and Déserti [Dés15] are respectively the main input for case 1), 4), 5), 6) in the classification.…”