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:



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> 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?
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By "generic singularities" I mean that $M$ is a simple polyhedron. See for instance [this paper of Alexander and Bishop](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.6959). 


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:



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



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


