I am from physics background so I apologize in advance if my question is trivial.


Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$ ([here](https://core.ac.uk/download/pdf/82773183.pdf)). I wonder if it is possible to work out an ideal triangulation for the manifold he constructs.


Some examples of hyperbolic 3-manifold with explicit ideal triangulation and with some simple non-trivial mapping class group would also be very helpful.



Ideal triangulation of hyperbolic 3-manifold with generic mapping class group
The hyperbolic manifolds considered by Kojima are closed, so do not admit ideal triangulations in the "usual sense". (They admit "partially flat, spun, ideal triangulations" but I am not sure that is what you are interested in.)


It is an open question whether or not every finite volume cusped hyperbolic three-manifold admits an ideal triangulation (with all tetrahedra positively oriented).


Finally, the standard example of a manifold admitting an ideal triangulation (with all tetrahedra positively oriented) is the figure-eight knot complement. This manifold, called m004 in the snappy census, has mapping class group isomorphic to $D\_4$. Taking normal covers of m004 gives examples with more symmetry.


See the book [Hyperbolic knot theory](https://bookstore.ams.org/cdn-1598077516917/gsm-209), by Jessica Purcell, as reference on this material.


