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.

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This work was supported by JSPS KAKENHI Grant Numbers 24224002, 15H05739.

Received 22 January 2018

Accepted 9 February 2019

Published 17 May 2022