# New PDF release: Some properties of the Cremona group

By Julie Déserti

Summary. We bear in mind a few homes, regrettably now not all, of the Cremona

group.

We first commence by way of proposing a pleasant evidence of the amalgamated product

structure of the well known subgroup of the Cremona workforce made of the

polynomial automorphisms of C2. Then we care for the type of

birational maps and a few purposes (Tits substitute, non-simplicity...)

Since any birational map might be written as a composition of quadratic

birational maps as much as an automorphism of the complicated projective plane,

we spend time on those targeted maps. a few questions of staff idea are

evoked: the class of the finite subgroups of the Cremona crew and

related difficulties, the outline of the automorphisms of the Cremona

group and the representations of a few lattices within the Cremona group.

The description of the centralizers of discrete dynamical platforms is an

important challenge in genuine and complicated dynamic, we describe the kingdom of

the artwork for this challenge within the Cremona group.

Let S be a compact advanced floor which consists of an automorphism f

of confident topological entropy. both the Kodaira size of S is zero

and f is conjugate to an automorphism at the special minimum version of

S that is both a torus, or a K3 floor, or an Enriques floor, or S is

a non-minimal rational floor and f is conjugate to a birational map of

the advanced projective aircraft. We take care of effects bought during this last

case: building of such automorphisms, dynamical homes (rotation

domains...).

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**Extra info for Some properties of the Cremona group**

**Sample text**

P2ν−1 . The corresponding Cremona transformation is a de Jonqui`eres transformation. 2 Here a rational fibration is a rational application from P2 (C) into P1 (C) whose fibers are rational curves. 3 The de Jonqui` eres group is birationally isomorphic to the subgroup of Bir(P1 (C) × P1 (C)) which preserves the first projection p : P1 (C) × P1 (C) → P1 (C). 38 Julie D´eserti Indeed let Γ be an element of Λf . Let Ξ be the pencil of curves of Λf that have in common with Γ a point m distinct from p1 , .

The length of the element given by Danilov is 26. ) is invariant by inner conjugacy, we can thus assume that f has minimal length in its conjugacy class. 8 ([94]). Let f be an element of Aut(C2 ). Assume that det jac f = 1 and that f has minimal length in its conjugacy class. • If f is non trivial and if ℓ(f ) ≤ 8, the normal subgroup generated by f coincides with the group of polynomial automorphisms f of C2 with det jac f = 1; • if f is generic6 and if ℓ(f ) ≥ 14, the normal subgroup generated by f is strictly contained in the subgroup f ∈ Aut(C2 ) det jac f = 1 of Aut(C2 ).

Let us denote by Eclat(S) the union of the surfaces endowed with a birational morphism π : S′ → S modulo the following equivalence relation: S ∋ p1 ∼ p2 ∈ S if and only if ε−1 2 ε1 sends p1 onto p2 and is a local isomorphism between a neighborhood of p1 and a neighborhood of p2 . A point of Eclat(S) corresponds either to a point of S, or to a point on an exceptional divisor of a blow-up of S etc. Any surface S′ which dominates S embeds into Eclat(S). Let us consider the free abelian group Ec(S) generated by the points of Eclat(S); we have a scalar product on Ec(S) (p, p)E = −1, (p, q) = 0 if p = q.

### Some properties of the Cremona group by Julie Déserti

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