2-dimensional surfaces with negative Euler characteristic have a pants decomposition which cuts the surface into pairs of pants. I'm curious if there is an analogue a dimension up.
A pair of pants has three 1-dimensional boundary components. In the Poincaré disk, a pair of pants resembles a triangle defined by three disjoint geodesics, which are those boundary components. Here is an example with a pair of pants highlighted on the Klein Quartic surface.
In $\mathbb{H}^3$, I'm picturing a volume with four boundary components (geodesic surfaces), something with a tetrahedral look in the Poincaré ball model.
If such a thing is a useful concept and has appeared in the literature, I would appreciate references.