Unions of 3-punctured spheres in hyperbolic 3-manifolds
We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyperbolic 3-manifold. Furthermore, we investigate bounds of the moduli of adjacent cusps for the union of linearly placed 3-punctured spheres.