Algebraic & Geometric Topology 10 (2010) 1565–1607

DOI: 10.2140/agt.2010.10.1565

Abstract

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–th pants curve is defined by a complex parameter τ_i in C. The associated holonomy representation ρ: π₁(Σ) → 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 π₁(Σ), are polynomials in the τ_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 ρ(γ), as polynomials in the τ_i, and the Dehn–Thurston coordinates of γ relative to 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 Σ as the bending measure tends to zero.

Keywords

Kleinian group, Dehn–Thurston coordinates, projective structure, plumbing construction, trace polynomial

Mathematical Subject Classification

Primary: 57M50

Secondary: 30F40

References

Publication

Received: 15 January 2010
Revised: 25 May 2010
Accepted: 1 June 2010
Published: 9 July 2010

Authors