Ratner’s theorem implies strong topological rigidity for immersed totally geodesic submanifolds in finite-volume locally symmetric spaces. In infinite-volume settings, however, the topological behavior of totally geodesic submanifolds is much less understood: while it has been studied in certain rank-one cases, the higher-rank setting has remained unexplored. In this talk, I will describe a construction of the first higher-rank examples exhibiting a failure of such rigidity using floating geodesic planes, joint work with Hee Oh. More precisely, we construct a Zariski-dense Hitchin surface group inside $\mathrm{SL}_3(\mathbb{Z})$ whose associated locally symmetric space contains floating geodesic planes whose closures have non-integer Hausdorff dimension.