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2010 Top terms of polynomial traces in Kra's plumbing construction
Sara Maloni, Caroline Series
Algebr. Geom. Topol. 10(3): 1565-1607 (2010). DOI: 10.2140/agt.2010.10.1565


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.


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Sara Maloni. Caroline Series. "Top terms of polynomial traces in Kra's plumbing construction." Algebr. Geom. Topol. 10 (3) 1565 - 1607, 2010.


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

zbMATH: 1268.57008
MathSciNet: MR2661536
Digital Object Identifier: 10.2140/agt.2010.10.1565

Primary: 57M50
Secondary: 30F40

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

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 3 • 2010
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