Top terms of polynomial traces in Kra’s plumbing construction

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 $ρ : π_{1} (Σ) \to PSL (2, ℂ)$ gives a projective structure on $Σ$ which depends holomorphically on the $τ_{i}$ . In particular, the traces of all elements $ρ (γ), γ \in π_{1} (Σ)$ , 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.

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Keywords

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

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 30F40

References

Publication

Received: 15 January 2010

Revised: 25 May 2010

Accepted: 1 June 2010

Published: 9 July 2010

Authors

Sara Maloni
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom
http://www.warwick.ac.uk/~marhal
Caroline Series
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom
http://www.warwick.ac.uk/~masbb/

Volume 10, issue 3 (2010)

Sara Maloni and Caroline Series

Abstract