Real spectrum and oriented Gromov equivariant compactifications of character varieties

The character variety $\mathrm{\Xi }$ of a finitely generated group $\mathrm{\Gamma }$ in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ has many compactifications. We construct a continuous surjection from the real spectrum compactification ${\mathrm{\Xi }}^{\mathrm{R}\mathrm{S}\mathrm{p}}$ to the oriented Gromov equivariant compactification. Our construction is based on a geometric interpretation of the elements of $\mathrm{\partial }{\mathrm{\Xi }}^{\mathrm{R}\mathrm{S}\mathrm{p}}$ as $\mathrm{\Gamma }$-actions by isometries on $\mathbb{R}$-trees. We endow these $\mathbb{R}$-trees with an orientation induced by the standard orientation on the circle, which we characterize by a semialgebraic equation. Moreover, we describe the $\mathrm{\Gamma }$-actions by orientation preserving isometries on oriented $\mathbb{R}$-trees, which arise in both compactifications, as limits of $\mathrm{\Gamma }$-actions on the oriented hyperbolic plane, via asymptotic cones endowed with an ultralimit orientation.