Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties

We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $Ξ\times Y \rightarrow Ξ$ extends to a $Γ$-equivariant continuous surjection $(Ξ\times Y)^{\mathrm{RSp}} \rightarrow Ξ^{\mathrm{RSp}}$. 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 $Y^{\mathrm{RSp}}$. 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.