I am reading Sullivan's 1985 Non-wandering paper(for the paper, see https://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf). Section 3 in the paper says that a maximal collection of disjoint simple closed geodesics decomposes a hyperbolic Riemann surface into components conformally equivalent to 'funnels' or '(degenerated) pair of pants'. It referred to Thurston's notes 'Geometry and topology of 3-manifolds', but I failed to find the proof. (I am very sorry to ask for help since the proof surely exists somewhere). Or are there any other recommendations for this proposition?
And here is another somewhat vague question. Can we represent Kleinian groups into some holonomy group of some 'Lorentian' manifolds? I am just wondering about the relationship between Kleinian groups and physics, although I know almost nothing about physics (except a little special relativity).
Thank you for your help in advance!
The only reference I know for the decomposition of a Riemann surface, compact or not, is the book of Hubbard, matrix editions, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1: Teichmüller Theory.