Let be a surface of negative Euler characteristic together with a pants decomposition . Kra’s plumbing construction endows with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or “plumb”, adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the –th pants curve is defined by a complex parameter . The associated holonomy representation gives a projective structure on which depends holomorphically on the . In particular, the traces of all elements , are polynomials in the .
Generalising results proved by Keen and the second author [Topology 32 (1993) 719–749; arXiv:0808.2119v1] and for the once and twice punctured torus respectively, we prove a formula giving a simple linear relationship between the coefficients of the top terms of , as polynomials in the , and the Dehn–Thurston coordinates of relative to .
This will be applied in a later paper by the first author to give a formula for the asymptotic directions of pleating rays in the Maskit embedding of as the bending measure tends to zero.
Sara Maloni. Caroline Series. "Top terms of polynomial traces in Kra's plumbing construction." Algebr. Geom. Topol. 10 (3) 1565 - 1607, 2010. https://doi.org/10.2140/agt.2010.10.1565