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.
Categories: math.GT math.AG math.GR