Markoff triples and quasifuchsian groups

B. H. Bowditch

We study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type-preserving representations of the once-punctured torus group into $ SL(2,{\Bbb C}) $. In particular, we explore which Markoff triples correspond to quasifuchsian representations. We derive a variation of McShane's identity for quasifuchsian groups. In the case of non-discrete representations, we attempt to relate the asymptotic behaviour of Markoff triples to the realisability of laminations in hyperbolic 3-space. We also consider how some of these issues are related for more general surfaces.

1991 Mathematics Subject Classification: 57M50

Keywords: Markoff triple, quasifuchsian group, punctured torus, representation space, lamination.

Proc. London Math. Soc. 77 (1998) 697-736.


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