Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 10, Number 3 (2010), 1565-1607.
Top terms of polynomial traces in Kra's plumbing construction
Let be a surface of negative Euler characteristic together with a pants decomposition . Kra’s plumbing construction endows 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 –th pants curve is defined by a complex parameter . The associated holonomy representation gives a projective structure on which depends holomorphically on the . In particular, the traces of all elements , are polynomials in the .
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 , as polynomials in the , and the Dehn–Thurston coordinates of relative to .
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 as the bending measure tends to zero.
Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1565-1607.
Received: 15 January 2010
Revised: 25 May 2010
Accepted: 1 June 2010
First available in Project Euclid: 19 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F40: Kleinian groups [See also 20H10]
Maloni, Sara; Series, Caroline. Top terms of polynomial traces in Kra's plumbing construction. Algebr. Geom. Topol. 10 (2010), no. 3, 1565--1607. doi:10.2140/agt.2010.10.1565. https://projecteuclid.org/euclid.agt/1513715145