Real spectrum and oriented Gromov equivariant compactifications of character varieties

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