Volume 10, issue 3 (2010)

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

Volume 17
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

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