Moduli spaces of surfaces

© 2014 Dr. Yi HuangWe give a generalisation of λ-length coordinates for theTeichmüller space of bordered hyperbolic surfaces, before defining moduli spaces and Teichmüller spaces for crowned surfaces and establishing mixed λ-length and Fenchel-Nielsen coordinates for the latter. We then define a mapping class group invariant Weil-Petersson 2-form and find a presentation for it in terms of mixed coordinates. We also prove McShane identities for crowned surfaces, closed surfaces with one marked point and quasi-Fuchsian representations of the thrice-punctured projective plane. Finally, we geometrically interpret Bowditch-type proofs of the McShane identity using the ideal Ptolemy relation