Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

Question

Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ideal vertices to get a triangulation of a manifold with torus boundary.

I am interested in the case where the link of an ideal vertex is instead a trivalent graph, so the truncation would give a triangulation of a manifold with a boundary component of genus $\ge 2$. Does this make sense combinatorially? Are there examples written down somewhere?

Answer 1

A nice class of examples are the (generalized) triangulations with only one edge. The manifolds obtained by removing an open neighborhood of the vertex have totally geodesic boundary (or a cusp in the genus $0$ case) with minimal volume for the Euler characteristic of the boundary. This was proved in a paper of Miyamoto.

Thurston's tripus manifold is an example of the genus 2 case.

Answer 2

In addition to the answers provided above, you might consider looking at

Heard, Damian; Hodgson, Craig; Martelli, Bruno; Petronio, Carlo, Hyperbolic graphs of small complexity, Exp. Math. 19, No. 2, 211-236 (2010). ZBL1207.57024.

In section 6, there is a table of examples of manifolds like those above with genus 2 boundary. The tables also give approximate computations of geometric data that we can associate to the manifolds.

Answer 3

I think you mean, in the first paragraph “the link of each ideal vertex is a torus” and in the second paragraph “the link of an ideal vertex is instead a surface of genus two”.

Such triangulations are sometimes called “pseudo-triangulations”, and the spaces they give “pseudo-manifolds”. For a recent discussion see the paper Traversing three-manifold triangulations and spines by Rubinstein, Segerman, and Tillmann.