Volume 10, issue 3 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 6, 3073–3719
Issue 5, 2459–3071
Issue 4, 1827–2458
Issue 3, 1253–1825
Issue 2, 621–1251
Issue 1, 1–620

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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