Top terms of polynomial traces in Kra’s plumbing construction

Maloni, Sara¹ ; Series, Caroline¹

¹Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1565-1607

Résumé

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 ∈ 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 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.

Détail

DOI : 10.2140/agt.2010.10.1565

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

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