Let \(F_{g,m}\) denote an orientable compact surface of genus \(g\) with \(m\) boundary curves and let \(F^{'}_{g,m}\) denote the surface \(F_{g,m}\) with one point \(p\) deleted. The fundamental group \(G^{'}_{g,m}\) of \(F^{'}_{g,m}\) is free on generators \[ a_{1}, b_{1}, \dots , a_{g}, b_{g}, c_{1}, \dots , c_{m}. \] Let \(\widehat{R}^{'}_{g,m}\) denote the space of all faithful representations \(\rho\) of \(G^{'}_{g,m}\) into \(SL(2, {\mathbb C})\) such that \(\rho(d)\) is parabolic and \(tr \rho(d) = -2.\) The group \(SL(2, {\mathbb C})\) acts on \(\widehat{R}^{'}_{g,m}\) and \(R^{'}_{g,m}\) - the orbit space.
The purpose of this paper is to give a system of coordinates called \(\lambda\)-lengths. They are complexifications of the \(\lambda\)-lengths introduced by R. C. Penner to parametrize the decorated Teichmüller space of punctured surface groups. A useful property of Penner’s \(\lambda\)-lengths is that they admit a rational representation of the mapping class group. Using the Ptolemy equation, the author proved that the mapping class group \(MC^{'}_{g,m}\) (fixing all boundary points) acts as a group of rational transformations on \({\mathbb C}^{6g-3+2m}.\) He also proves that any mapping class which is a composite of Dehn twists on mutually disjoint simple loops has no fixed points in \(R^{'}_{g,m}.\) This fact is applied to find hyperbolic 3-manifolds which fibre over the circle.