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 $\P$. 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 $τ_i \in \C$. The associated holonomy representation $ρ: π_1(Σ) \to PSL(2,\C)$ 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.

42 pages, 17 figures

Cited by in corpus (1)

• The Maskit embedding of the twice punctured torus