Markoff triples and quasifuchsian groups

1998/11/01 by BH Bowditch, Β. H. Bowditch · 7 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · doi:10.1112/s0024611598000604

Abstract

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 SL(2,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.

Cited by

• McShane-Rivin norm balls and simple-length multiplicities
• The modular action on PSL2(R)-characters in genus 2
• Limit cones of multi-Fuchsian representations
• Bowditch representations in Gromov-hyperbolic spaces : characterizations, dynamics of Out(\mathbbF₂) and recognition
• An elementary proof of uniqueness of Markoff numbers which are prime powers
• Primitive stability and the Q-conditions for the rank two free group in hyperbolic d-space
• Short geodesics and end invariants

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