Two-dimensional Cremona groups acting on simplicial complexes

Abstract:

We show that the $2$-dimensional Cremona group \[ \operatorname {Cr}_2 = \operatorname {Aut}_k\;k(X,Y)\] 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.