Top terms of polynomial traces in Kra's plumbing construction

Abstract

Let $Σ$ be a surface of negative Euler characteristic together with a pants decomposition $¶$. Kra's plumbing construction endows $Σ$ 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 $τ_i \in \C$. The associated holonomy representation $ρ: π_1(Σ) \to PSL(2,\C)$ gives a projective structure on $Σ$ which depends holomorphically on the $τ_i$. In particular, the traces of all elements $ρ(γ), γ\in π_1(Σ)$, are polynomials in the $τ_i$. 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 $ρ(γ)$, as polynomials in the $τ_i$, and the Dehn-Thurston coordinates of $γ$ relative to $¶$. This will be applied elsewhere to give a formula for the asymptotic directions of pleating rays in the Maskit embedding of $Σ$ as the bending measure tends to zero.