While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an affirmative result in this direction. One higher-dimensional generalization of a geodesic line is a totally geodesic immersed submanifold, and for these, Nimish Shah has proved (*Closures of totally geodesic immersions in manifolds of constant negative curvature*) that similarly to what happens on a flat torus, the closure of a complete immersed totally geodesic submanifold *of dimension at least $2$* in any compact hyperbolic manifold $M$ is always a totally geodesic immersed submanifold of $M$. (I think this result has subsequently been extended to cover the weaker assumption that $M$ is complete, not necessarily compact.)

In moduli space $\mathcal{M}_g$ with the Teichmuller metric, where again the closure of a geodesic can take a wild fractal shape, Mirzakhani and her coworkers have proved that the closure of a *complex* geodesic is always an algebraic subvariety. This setting, while technically rather more involved, has been frequently compared to the homogeneous one of the first paragraph. In light of this, here is my question, which I will split into two variants as I am unsure which of them (if either) makes more sense:

For a totally geodesic immersed submanifold $N \to \mathcal{M}_g$ of dimension at least $2$, is it reasonable to expect that the closure is still an immersed submanifold?

It is easy to imagine a definition of a higher-dimensional

*complex*totally geodesic immersed submanifold in $\mathcal{M}_g$. Extending Mirzakhani et. al.'s theorem, can it be shown that the closure of such an immersed complex submanifold is an algebraic subvariety?