Top terms of polynomial traces in Kra's plumbing construction

Abstract

Let $\Sigma$ be a surface of negative Euler characteristic together with a pants decomposition $\mathcal{P}$. 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 $\rho :{\pi }_1\left(\Sigma \right)\to PSL\left(2,ℂ\right)$ gives a projective structure on $\Sigma$ which depends holomorphically on the ${\tau }_i$. In particular, the traces of all elements $\rho \left(\gamma \right),\gamma \in {\pi }_1\left(\Sigma \right)$, are polynomials in the ${\tau }_i$.

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 $\rho \left(\gamma \right)$, as polynomials in the ${\tau }_i$, and the Dehn–Thurston coordinates of $\gamma$ relative to $\mathcal{P}$.

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 $\Sigma$ as the bending measure tends to zero.