Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties

We construct universal geometric spaces over the real spectrum compactification ${\mathrm{\Xi }}^{\mathrm{R}\mathrm{S}\mathrm{p}}$ of the character variety $\mathrm{\Xi }$ of a finitely generated group $\mathrm{\Gamma }$ in ${\mathrm{S}\mathrm{L}}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which ${\mathrm{S}\mathrm{L}}_n(\mathbb{R})$ acts by algebraic automorphisms (such as ${\mathbb{P}}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of ${\mathrm{S}\mathrm{L}}_n(\mathbb{R})$), the projection map $\mathrm{\Xi }\times Y\to \mathrm{\Xi }$ extends to a $\mathrm{\Gamma }$-equivariant continuous surjection $(\mathrm{\Xi }\times Y{)}^{\mathrm{R}\mathrm{S}\mathrm{p}}\to {\mathrm{\Xi }}^{\mathrm{R}\mathrm{S}\mathrm{p}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{R}\mathrm{S}\mathrm{p}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y={\mathbb{P}}^1$, the image is a real subtree.