Mathematics – General Topology
Scientific paper
2010-01-14
Mathematics
General Topology
42 pages, 17 figures
Scientific paper
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.
Maloni Sara
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