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Top terms of polynomial traces in Kra's plumbing construction

Sara Maloni and Caroline Series

Algebraic & Geometric Topology 10 (2010) 1565–1607

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 τi . The associated holonomy representation ρ: π1(Σ) PSL(2, ) 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 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 τi, 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.

Kleinian group, Dehn–Thurston coordinates, projective structure, plumbing construction, trace polynomial
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 30F40
Received: 15 January 2010
Revised: 25 May 2010
Accepted: 1 June 2010
Published: 9 July 2010
Sara Maloni
Mathematics Institute
University of Warwick
United Kingdom
Caroline Series
Mathematics Institute
University of Warwick
United Kingdom