Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Question

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing geodesics.

I will start with a general question:

Is it true that for generic metrics on $M$ and generic points $p$, the cut locus of $p$ is a $(n-1)$-dimensional polyhedron with generic singularities?

By "generic singularities" I mean that $M$ is a simple polyhedron. See for instance this paper of Alexander and Bishop.

This property is certainly not satisfied for some important specific metrics: for instance, if $M$ is a round sphere the cut locus is a point, no matter where $p$ is. If $M$ is a flat torus, we get a generic polyhedron for generic flat metrics. What about hyperbolic manifolds? So, this is my question:

Let $M^n$ be a hyperbolic $n$-manifold. Is the cut locus of a generic point a $(n-1)$-polyhedron with generic singularities?

Of course I am mostly interested in the case $n=3$. In dimension $n=2$ one may also pick a generic hyperbolic metric.

Edit: In dimension 1, a simple polyhedron is a graph with vertices of valence 2 or 3. In dimension 2, it is a polyhedron such that the link of a point is either a circle, a circle with a diameter, or a circle with three radii.

In general, a $n$-dimensional compact polyhedron is simple if every point has a neighborhood which is the cone over the $(k-1)$-skeleton of the $(k+1)$-simplex, times a $(n-k)$-disc.

Answer 1

The question you pose is stated as an open question (in the 3-dimensional hyperbolic case) in the following paper:

Díaz, Raquel; Ushijima, Akira On the properness of some algebraic equations appearing in Fuchsian groups. Topology Proc. 33 (2009), 81–106.

Quoting from the review on mathscinet:

[the paper] takes its motivation from the fact that, apparently, the statement about the genericity of Dirichlet fundamental polyhedra is open for $\mathbb{H}^3$. (According to the authors, the paper of T. Jørgensen and A. Marden [in Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), 69--85, Springer, New York, 1988] has a gap which the present authors have so far been unable to fix.)

Answer 2

As indicated by Mohan, Buchner has written some papers on the subject in the 70s, for instance here. In his study, he fixes the manifold $M$ and the point $p$, and he varies the metrics $g$. He defines a notion of cut-stable metric. Cut-stable metrics form a dense open subset of the set of all riemannian metrics.

The cut locus $C(g)$ determined by a cut-stable metric $g$ is locally stable (it is the same PL object as one varies $g$ locally). Moreover, the local structure of $C(g)$ is indeed generic when $M$ has low dimensions: however the "generic polyhedra" one can obtain are a strictly larger class than the "simple polyhedra" I defined above.

For instance, if $M^2$ is a surface the cut-locus $C(g)$ of a cut-stable metric $g$ is a graph with vertices having valence 1, 2, or 3. Vertices with valence 1 are inavoidable for instance on a 2-sphere, for obvious reasons (the cut locus is a tree). However a simple polyhedron contains only vertices of valence 2 and 3.

Analogously, if $M^3$ is a 3-manifold the cut-locus $C(g)$ of a cut-stable metric $g$ is a 2-dimensional polyhedron whose local structure belongs to finitely many types. There are five links one can get: the three arising in the definition of a simple polyhedron (circle, circle with diameter, circle with three radii), plus two more (segment and a circle with a radius). Therefore the answer to my first question in low dimensions is:

Let $M^n$ has dimension 2 or 3. For generic metrics on $M$ and generic points $p$, the cut locus of $p$ is a (n−1)-dimensional polyhedron with generic singularities. However, the polyhedron may not be simple.

On the other hand, if the cut-stable metric $g$ has non-positive curvature, there are no conjugate points and the non-simple singularities cannot arise: thus we really get a simple polyhedron in this case (at least in dimension $n=2$ and $n=3$).

In all this discussion the metric $g$ is generic, so it gives no information for hyperbolic 3-manifolds (i.e. the second question).

Answer 3

If curvature $\le -1$ then cut locus is is glued from the boundary of fundamental domain of $\pi_1$-action on the universal cover. If curvature $=-1$ then the fundamental domain is a convex polyhedral. The gluing maps are piecewise isometries, so the result is $(n-1)$-polyhedron, But one construct an action which gives a complicated link. Take $\mathbb Z^2$ action on hyperbolic $3$-space such that it has one fixed point on the absolute and action on its horosphere is standard action of $\mathbb Z^2$ on the Euclidean plane. (I do not see how to make a compact example.)

For the first question, in some sense the answer is "YES" if $\dim \ge 3$. I.e. there is a $G_\delta$-set in $C^\infty$-topology which satisfies your condition and dense in Gromov--Hausdorff metric.