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Transactions of the American Mathematical Society

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Two-dimensional Cremona groups acting on simplicial complexes


Author: David Wright
Journal: Trans. Amer. Math. Soc. 331 (1992), 281-300
MSC: Primary 14E07; Secondary 14J50, 20F05
MathSciNet review: 1038019
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Abstract: We show that the $ 2$-dimensional Cremona group

$\displaystyle \operatorname{Cr}_2 = \operatorname{Aut}_k\;k(X,Y)$

acts on a $ 2$-dimensional simplicial complex $ C$, which has as vertices certain models in the function field $ k(X,Y)$. The fundamental domain consists of one face $ F$. This yields a structural description of $ \operatorname{Cr}_2$ as an amalgamation of three subgroups along pairwise intersections. The subgroup $ {\text{GA}}_2 = \operatorname{Aut}_k\;k[X,Y]$ (integral Cremona group) acts on $ C$ by restriction. The face $ F$ has an edge $ E$ such that the $ {\text{GA}}_2$ translates of $ E$ form a tree $ T$. The action of $ {\text{GA}}_2$ on $ T$ yields the well-known structure theory for $ {\text{GA}}_2$ as an amalgamated free product, using Serre's theory of groups acting on trees.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1038019-2
Article copyright: © Copyright 1992 American Mathematical Society