Continuous automorphisms of Cremona groups

Urech, Christian; Zimmermann, Susanna

doi:10.1142/s0129167x21500191

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…Finally, in [UZ21], the authors prove that any homeomorphism of Cr 3 (k), with respect to either the Zariski or the Euclidean topology, is a composition of an inner and a field automorphism for k = R or C. Thus our examples constitutes, to our knowledge, the first examples of non-continuous automorphisms of Cr 3 (C).…”

Section: Non-generation Of Aut(cr 3 (C)) By Inner and Field Automorph...mentioning

confidence: 67%

Rigid birational involutions of ℙ 3 and cubic threefolds

Zikas¹

2023

Journal De L’École Polytechnique — Mathématiques

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We construct families of birational involutions on P 3 or on a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.Résumé (Involutions birationnelles rigides de P 3 et de cubiques lisses de dimension 3)Nous construisons des familles d'involutions birationnelles sur P 3 ou sur une cubique lisse de dimension 3 qui ne s'intègrent pas dans une relation élémentaire non triviale de liens de Sarkisov. En conséquence, nous construisons de nouveaux homomorphismes à partir de leur groupe de transformations birationnelles, redémontrant de manière effective leur non-simplicité. Nous prouvons également que ces groupes admettent une structure de produit libre. Enfin, nous produisons des automorphismes de ces groupes qui ne sont pas engendrés par des automorphismes intérieurs et des automorphismes de corps.

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Section: Non-generation Of Aut(cr 3 (C)) By Inner and Field Automorph...mentioning

confidence: 67%

Rigid birational involutions of ℙ 3 and cubic threefolds

Zikas¹

2023

Journal De L’École Polytechnique — Mathématiques

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mentioning

confidence: 66%

“…For Y = P 3 and any ρ as above, the group automorphism φ(ρ) : Cr 3 (C) → Cr 3 (C) is not a homeomorphism with respect to either the Zariski or the Euclidian topology on Cr 3 (C). Indeed by the results of [UZ21], any homeomorphism of Cr 3 (C), with respect to either of the two topologies, is the composition of a field automorphism with an inner automorphism.…”

Section: Consequencesmentioning

confidence: 99%

Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds

Zikas¹

2021

Preprint

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We construct families of birational involutions on P 3 or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.

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“…In fact,

\operatorname{Bir}(\mathbb {P}^3_\mathbb {C})

is not even generated by its algebraic subgroups [2, Theorem C]. The Euclidean topology on Cremona groups is largely unstudied and results can be found in [1, 3, 16, 18].…”

Section: Introductionmentioning

confidence: 99%

Properties of the Cremona group endowed with the Euclidean topology

Bergner¹,

Zimmermann

2023

Bulletin of London Math Soc

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Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that Cremona groups do not contain any non-stationary sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.

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Continuous automorphisms of Cremona groups

Cited by 5 publications

References 19 publications

Rigid birational involutions of ℙ 3 and cubic threefolds

Rigid birational involutions of ℙ 3 and cubic threefolds

Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds

Properties of the Cremona group endowed with the Euclidean topology

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