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 $\hat{I}“<{\mathrm{SL}}_3(\mathbb{R})$ such that the Hitchin manifold $\hat{I}“\backslash {\mathrm{SL}}_3(\mathbb{R})/\mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $\hat{I}“$ can be chosen inside ${\mathrm{SL}}_3(\mathbb{Z})$.

New title, 47 pages, 7 figures