, Let f n ? · · · ? f 1 = id P 2 be a relation in Bir R (P 2 ) with f i ? G * ? G ? . By Theorem 2.5(1) for each i = 1, . . . , r there are links ? i1, ? iri such that f i = ? iri ? · · · ? ? i1 . Then ? nrn ? · · · ? ? n1 ? · · · ? ? 1r1 ? · · · ? ? 11 = id P
, Remark 3.10), and each elementary relation corresponds to a loop around the boundary of an elementary disc (Theorem 2.5). Elementary discs are classified in Proposition 2.8. The boundary of discs D 1 , D 3 and D 5 respectively corresponds to a relation in G ? ? G * , in G ? and in G * , respectively. We attach to *, is a relation inside the groupoid Sar R (P 2 ) and is thus a composition of conjugates of elementary relations
,
, Lemma 3.6 and Corollary 3.12. So the amamlgamated structure on Bir R (P 2 ) is nontrivial. Finally, the index of G * is uncountable by Lemma 3.7 and the index of G ? is uncountable by Corollary 3.12. Theorem 1.3. The homomorphism ? : Bir R (P 2 ) ? (0,1] Z/2Z from Proposition 3.11 coincides with the one given in [18, Proposition 5.3] since its restriction to J ? is the surjective homomorphism ? : J ? ? (0,1] Z/2Z and its kernel contains G * by construction, hence it also contains Aut R (P 2 ) and J * . The kernel of ? is, vol.3
, Then every element of Bir R (P 2 ) of finite order has a fixed point on T . It follows that very finite subgroup of Bir R (P 2 ) has a fixed point on T [15, §I.6.5, Corollary 3], and is in particular conjugate to a subgroup of G * or of G ? . For infinite algebraic subgroups of Bir R (P 2 ), it suffices to check the claim for the maximal algebraic subgroups of Bir R (P 2 )
, ) X is a del Pezzo surface of degree 6 with a birational morphism X ?? F 0 blowing-up a pair of non-real conjugate points
, Simple relations in the Cremona group, Proc. Amer. Math. Soc, vol.140, issue.5, pp.1495-1500, 2012.
, Quotients of higher dimensional cremona groups, p.14, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01981369
, Cremona groups of real surfaces, Automorphisms in birational and affine geometry, vol.79, p.13, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00838522
, Normal subgroups in the Cremona group, Acta Math, vol.210, issue.1, pp.31-94, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00705299
, Le trasformazioni generatrici del gruppo cremoniano nel piano, pp.861-874, 1901.
, Generators in the two-dimensional Cremona group over a nonclosed field. Translation of the 1991 paper from Trudy Mat, Inst. Steklov, p.14, 1991.
, Proof of a theorem on relations in the two-dimensional Cremona group. Uspekhi Mat. Nauk, vol.40, pp.255-256, 1985.
, Factorization of birational mappings of rational surfaces from the point of view of Mori theory, vol.51, p.6, 1996.
, Relations in the Sarkisov program, Compos. Math, vol.149, issue.10, pp.1685-1709, 2013.
, Groupes de transformations birationnelles de surfaces. Mémoire d'habilitation à diriger des recherches, 2010.
, Signature morphisms from the cremona group over a non-closed field, vol.7, p.14, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01719067
, Infinite algebraic subgroups of the real cremona group, Osaka J. of Math, vol.55, issue.4, p.16, 2018.
, Birational diffeomorphisms of the real projective plane, Comment. Math. Helv, vol.80, issue.3, pp.517-540, 2005.
, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math, vol.327, issue.2, pp.12-80, 1981.
, Trees. Springer Monographs in Mathematics, 2003.
, Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc, vol.331, issue.1, pp.281-300, 1992.
, Subgroups of odd order in the real plane Cremona group, J. Algebra, vol.461, issue.2, pp.87-120, 2016.
, The abelianisation of the real Cremona group, vol.13, p.15, 2009.
, Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d'Angers, 49045 Angers Cedex 1, France E-mail address: susanna.zimmermann@univ-angers