  Let $\Sigma$ be a surface of negative Euler characteristic together with a
pants decomposition $\P$. Kra's plumbing construction endows $\Sigma$ 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 $i^{th}$ pants curve is
defined by a complex parameter $\tau_i \in \C$. The associated holonomy
representation $\rho: \pi_1(\Sigma) \to PSL(2,\C)$ gives a projective structure
on $\Sigma$ which depends holomorphically on the $\tau_i$. In particular, the
traces of all elements $\rho(\gamma), \gamma \in \pi_1(\Sigma)$, are
polynomials in the $\tau_i$.
  Generalising results proved in previous papers 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 $\rho(\gamma)$, as
polynomials in the $\tau_i$, and the Dehn-Thurston coordinates of $\gamma$
relative to $\P$.
  This will be applied elsewhere to give a formula for the asymptotic
directions of pleating rays in the Maskit embedding of $\Sigma$ as the bending
measure tends to zero.

General Topology;Geometric Topology