Ideal triangulation of compact 3-manifolds So I know that the figure-8 knot complement decompose into two ideal tetrahedra. But as far as I‘m concerned the knot complement is a non-compact 3-manifold.
Is there an easy example for an ideal triangulation of a compact 3-manifold that admits a hyperbolic structure like the figure-8 knot complement? Is that even possible?
And are there any general theorems about triangulation like „All compact 3-manifolds decompose into tetrahedra…“?
I try to get an overview about this topic.
 A: The theorem that every 3-manifold $M$ can be triangulated using "ordinary" tetrahedra is Moise's Theorem; the wikipedia link has a reference to Moise's book.
The concept of an ideal triangulation is less standard than a triangulation, and your post did not offer a definition or a link to a definition. One definition that works for some interesting purposes is that an ideal triangulation is what you get from a simplicial complex by removing a subcomplex (a good example occurs on the outer space of a finite rank free group). With this definition, if the given complex is connected and the removed subcomplex is proper and nonempty then the resulting space is noncompact. It follows that you cannot put an ideal triangulation on any compact topological space whatsoever, unless that "ideal" triangulation is an actual triangulation.
However, it turns out that by relaxing the definition of ideal triangulation you do get some interesting constructions. If $M$ is a compact 3-manifold equipped with a hyperbolic structure then William Thurston constructed a kind of "ideal triangulation" of $M$ which is actually an ideal triangulation (in the previous, stricter sense) of the complement $M-C$ of a simple closed geodesic $C \subset M$; this ideal triangulation has the additional property that the removed subcomplex is a single vertex, one per component of $\partial M$. This construction is part of Thurston's Dehn surgery theorem (found in chapter 4). A closely related and rather general statement is that for any compact 3-manifold with boundary $M$, such that $\partial M$ is a union of tori, the interior $M - \partial M$ has an ideal triangulation (see these lecture notes by Alex Casella) in which the removed subcomplex is a finite set of vertices. Nonetheless, despite the utility and beauty of this construction, you cannot say that it gives an ideal triangulation of $M$ itself.
