Fractal closures of geodesic planes in Hitchin manifolds

Abstract

Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained elusive. We construct the first such examples using \emph{floating geodesic planes}. Specifically, we exhibit a Zariski-dense Hitchin surface group $\Gamma < \mathrm{SL}_3(\mathbb{R})$ such that the Hitchin manifold $\Gamma \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $\Gamma$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$.