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Subgroups of cremona groups

Urech, Christian. Subgroups of cremona groups. 2017, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_12384

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Abstract

The Cremona group in n-variables Cr_n(C) is the group of birational transformations of the complex projective n-space. This thesis contributes to the research on Cremona groups through the study of certain classes of „large'' subgroups. In the first part we consider algebraic embeddings of Cr_2(C) into Cr_n(C). In particular, we describe geometrical properties of an embedding of Cr_2(C) into Cr_5(C) that was discovered by Gizatullin. We also classify all algebraic embeddings from Cr_2(C) into Cr_3(C), and we partially generalize this result to embeddings of Cr_n(C) into Cr_{n+1}(C). In a second part, we look at degree sequences of birational transformations of varieties over arbitrary fields. We show that there exist only countably many such sequences and we give new obstructions on the degree growth of automorphisms of affine n-space. In the third part, we classify subgroups of Cr_2(C) containing only elliptic elements, i.e. elements whose iterates are of bounded degree. From this we deduce in particular the Tits alternative for arbitrary subgroups of Cr_2(C). In the last part, we show that every finitely generated simple subgroup of Cr_2(C) is finite and, under the hypothesis of an unproven conjectural lemma, that a simple group can be embedded into Cr_2(C) if and only if it can be embedded into PGL_3(C).
Advisors:Blanc, Jérémy and Cantat, Serge and Cheltsov, Ivan and Brion, Michel
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Algebra (Blanc)
UniBasel Contributors:Blanc, Jérémy
Item Type:Thesis
Thesis Subtype:Doctoral Thesis
Thesis no:12384
Thesis status:Complete
Bibsysno:Link to catalogue
Number of Pages:1 Online-Ressource (viii, 171 Seiten)
Language:English
Identification Number:
Last Modified:22 Apr 2018 04:32
Deposited On:04 Dec 2017 13:38

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