Top Terms of Polynomial Traces in Kra’s Plumbing Construction

Let Σ be a surface of negative Euler characteristic together with a pants decomposition P. 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 pants curve is defined by a complex parameter τi ∈ C. The associated holonomy representation ρ : π1(Σ) −→ PSL(2,C) gives a projective structure on Σ which depends holomorphically on the τi. In particular, the traces of all elements ρ(γ), γ ∈ π1(Σ), are polynomials in the τi. Generalising results proved in [5, 16] 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 τi, and the Dehn–Thurston coordinates of γ relative to P. 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, see [9]. MSC classification: 30F40, 57M50