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Top terms of polynomial traces in Kra's plumbing construction
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Maloni, Sara and Series, Caroline (2010) Top terms of polynomial traces in Kra's plumbing construction. Algebraic & Geometric Topology, Vol.10 (No.3). pp. 1565-1607. doi:10.2140/agt.2010.10.1565 ISSN 1472-2747.
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Official URL: http://dx.doi.org/10.2140/agt.2010.10.1565
Abstract
Let Sigma be a surface of negative Euler characteristic together with a pants decomposition P. Kra's plumbing construction endows Sigma 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 tau(i) is an element of C. The associated holonomy representation rho: pi(1) (Sigma) -> PSL(2, C) gives a projective structure on Sigma which depends holomorphically on the tau(i). In particular, the traces of all elements rho(gamma), gamma is an element of pi(1)(Sigma) are polynomials in the tau(i).
Generalising results proved by Keen and the second author [4; 13] 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 rho(gamma), as polynomials in the tau(i), and the Dehn-Thurston coordinates of gamma relative to P.
This will be applied in a later paper [7] by the first author to give a formula for the asymptotic directions of pleating rays in the Maskit embedding of Sigma as the bending measure tends to zero.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Kleinian groups, Euler characteristic, Polynomials | ||||
Journal or Publication Title: | Algebraic & Geometric Topology | ||||
Publisher: | Mathematical Sciences Publishers | ||||
ISSN: | 1472-2747 | ||||
Official Date: | 2010 | ||||
Dates: |
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Volume: | Vol.10 | ||||
Number: | No.3 | ||||
Page Range: | pp. 1565-1607 | ||||
DOI: | 10.2140/agt.2010.10.1565 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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