## Algebraic & Geometric Topology

### Top terms of polynomial traces in Kra's plumbing construction

#### Abstract

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 $τi∈ℂ$. The associated holonomy representation $ρ:π1(Σ)→ PSL(2,ℂ)$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1565-1607.

Dates
Revised: 25 May 2010
Accepted: 1 June 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715145

Digital Object Identifier
doi:10.2140/agt.2010.10.1565

Mathematical Reviews number (MathSciNet)
MR2661536

Zentralblatt MATH identifier
1268.57008

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

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

#### References

• L Bers, Inequalities for finitely generated Kleinian groups, J. Analyse Math. 18 (1967) 23–41
• M Dehn, Lecture notes from Breslau, Archives of the University of Texas at Austin (1922)
• A Fathi, F Laudenbach, V Poenaru (editors), Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France, Paris (1979) Séminaire Orsay, With an English summary
• L Keen, C Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993) 719–749
• I Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990) 499–578
• F Luo, Simple loops on surfaces and their intersection numbers, Math. Res. Lett. 5 (1998) 47–56
• S Maloni, The Maskit embedding of a hyperbolic surface, in preparation
• A Marden, Outer circles. An introduction to hyperbolic $3$–manifolds, Cambridge Univ. Press (2007)
• B Maskit, Moduli of marked Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974) 773–777
• K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York (1998)
• D Mumford, C Series, D Wright, Indra's pearls. The vision of Felix Klein, Cambridge Univ. Press, New York (2002)
• R C Penner, J L Harer, Combinatorics of train tracks, Annals of Math. Studies 125, Princeton Univ. Press (1992)
• C Series, The Maskit embedding of the twice punctured torus
• D P Thurston, On geometric intersection of curves on surfaces Available at \setbox0\makeatletter\@url http://www.math.columbia.edu/~dpt/ {\unhbox0