Action de groupe sur la compactification hybride

Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ be an algebraic group acting on $X$ whose action is closed. J. Poineau defined a compactification $X^\urcorner$ of $X(\mathbb{C})$ by using hybrid Berkovich spaces. We will focus on the extension of the action of $G$ on this compactification by characterising the set $\mathcal{U} \subset X^\urcorner$ where the action is well defined. We will also show that the quotient of $\mathcal{U}$ by the action of $G$ is homeomorphic to $(X/G)^\urcorner$, the compactification of $(X/G)(\mathbb{C})$. We then apply these results to $X = \mathrm{Rat}_d$, the space of rational maps and $G = \mathrm{SL}_2$. It gives the results of C. Favre-C. Gong in a more general setting. Furthermore, we get a compactification of $\mathrm{M}_d = \mathrm{Rat}_d/\mathrm{SL}_2$ where the boundary is made of orbits of non-archimedean rational maps. The results still holds if $\mathbb{C}$ is replaced by $k$ a non-trivially valued field and complex analytic spaces by Berkovich spaces over $k$ or if $X$ is the set of stable points of a $k$-variety defined in the sense of GIT.