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 punctured torus group into $\mbox{SL}(2,{\Bbb C})$. 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 might be related for more general surfaces.

1991 Mathematics Subject Classification: 57M50.