Abstract: We give a simple set of generators and relations for the Cremona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonquières group with the group of automorphisms of the plane, divided by one relation which is , where and .
V. A. Iskovskikh, Generators and relations in a two-dimensional Cremona group, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 5, 43-48. English transl. in Moscow Univ. Math. Bull. 38 (1983), no. 5, 56-63. MR 722450 (85e:14019)
V. A. Iskovskikh, Proof of a theorem on relations in the two-dimensional Cremona group, Uspekhi Mat. Nauk 40 (1985), no. 5 (245), 255-256. English transl. in Russian Math. Surveys 40 (1985), no. 5, 231-232. MR 810819 (87d:14007)