Algebraic & Geometric Topology

Top terms of polynomial traces in Kra's plumbing construction

Sara Maloni and Caroline Series

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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),γπ1(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.

Article information

Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1565-1607.

Received: 15 January 2010
Revised: 25 May 2010
Accepted: 1 June 2010
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F40: Kleinian groups [See also 20H10]

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


Maloni, Sara; Series, Caroline. Top terms of polynomial traces in Kra's plumbing construction. Algebr. Geom. Topol. 10 (2010), no. 3, 1565--1607. doi:10.2140/agt.2010.10.1565.

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