On the conjugacy problem for finite groups in the plane Cremona group

Abstract

We give a geometric condition for two finite subgroups $G\simeq G^{\prime }$ of the plane Cremona group ${\mathrm{Cr}}_2\left(C\right)$ to be conjugate. The argument is based on the properties of a “Burnside-type” ring $\Omega (P^2,G)$ encoding rational G-surfaces.