Fractal failures of Ratner rigidity in higher rank geometry

Abstract: Ratner's theorem shows that in a locally symmetric space of noncompact type and finite volume, every immersed totally geodesic subspace of noncompact type is topologically rigid: its closure is an immersed submanifold. We construct the first explicit higher-rank, infinite-volume examples in which this rigidity fails, via floating geodesic planes. Specifically, we exhibit a Zariski-dense Hitchin surface group $\Gamma<\mathrm{SL}_3(\mathbb{R})$ such that $$\Gamma\backslash \mathrm{SL}_3(\mathbb{R})/ \mathrm{SO}(3)$$ contains a sequence of immersed floating geodesic planes with fractal closures whose Hausdorff dimensions, non-integral, accumulate at $2$. Moreover, $\Gamma$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$. Our method uses Goldman's bulging deformations, but in higher rank new difficulties arise: unlike in rank one, where geodesics orthogonal to a hyperplane always diverge, here one must analyze the collective behavior of entire families of parallel geodesics inside flats under bulging, a phenomenon intrinsic to higher rank.