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:

  1. 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?

  2. 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?

  • $\begingroup$ Could you provide more precise references? $\endgroup$ – YCor Jun 14 '19 at 7:05
  • $\begingroup$ @YCor: Yes, I will add them shortly. Sorry about the delay with turning to your comment/request. $\endgroup$ – Vesselin Dimitrov Jul 8 '19 at 16:30

All of these results are about the dynamics of group actions.

There is no group action on the hyperbolic manifold itself, rather there is a group action on the frame bundle, which is a homogeneous space G/Gamma. Here G is a Lie group, and Gamma is a lattice. For hyperbolic n-manifolds, G is the isometry group of the n dimensional hyperbolic plane.

In the homogenous setting, Ratner's Theorem classifies orbit closures for homogeneous group actions, when the group is generated by unipotents. SL(2,R) is generated by unipotents, but the one parameter subgroup giving rise to geodesic flow is not. The orbits of SL(2,R) in G/Gamma project to totally geodesic planes in the hyperbolic manifold.

For hyperbolic manifolds, since the dynamics takes place on G/Gamma, there are many groups which act. Indeed, any Lie subgroup of G acts naturally on G/Gamma.

For moduli space, the situation is partially analogous, but also substantively different. The Hodge bundle over M_g, i.e. the set of pairs (X,omega) where X is a Riemann surface and omega is a holomorphic one form on X, admits a SL(2,R) action, even though M_g itself doesn't admit any such action. So the Hodge bundle is the analogue of G/Gamma, and indeed all of the work of Eskin-Mirzakhani-Mohammadi takes place on the Hodge bundle, using the dynamics of the SL(2,R) action.

The difference is that since SL(2,R) is a small group, there are very few subgroups that can act. For the diagonal subgroup, orbit closures can be complicated fractal objects. Orbit closures of unipotent flow, unlike in the homogenous case, are not currently understood. Orbit closures for all of SL(2,R), and for the upper triangular subgroup of SL(2,R), were studied by Eskin-Mirzakhani-Mohammadi.

The connection to your question is the SL(2,R) orbits map to complex geodesics.

Since the group GL(2,R) is so small, there are no other actions we can consider. Other totally geodesic submanifolds of M_g are not, as far as I know, tied to a group action with well understood dynamics.

Let me end with two side notes.

1) Stergios Antonakoudis has shown that the situation for complex disks is quite special; other bounded symmetric domains don't admit isometric maps to M_g: http://www.math.harvard.edu/~stergios/papers/TeichBSD.pdf.

2) I wrote a short piece for the Bulletin of the AMS introducing the work of Eskin-Mirzakhani-Mohammadi from a more elementary perspective: http://web.stanford.edu/~amwright/BilliardsToModuli.pdf.

  • $\begingroup$ Thank you for your answer and references! Perhaps the statements could still be true for different reasons not directly related to group actions? $\endgroup$ – Vesselin Dimitrov Jul 30 '15 at 23:50

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