The Markoff equation \(x^2 + y^2 + z^2 = 3 xyz\) is well known in diophantine approximation. Its solutions in natural numbers lead to the initial segment of the Markoff spectrum thus to those real numbers which are most poorly approximated by rationals.
H. Cohn [Ann. Math. (2) 61, 1-12 (1955; Zbl 0064.04303)] recognized the similarity between the Markoff equation and one used by R. Fricke to parametrize the space of hyperbolic once-punctured tori. (The equations differ only in the right hand side coefficient.) Cohn showed that the Markoff equation can be seen as directly related to the traces of generators of the commutator subgroup of the modular group \(\text{PSL}(2, {\mathbb{Z}})\). This subgroup is isomorphic to the free group on two letters and uniformizes a particular hyperbolic once-punctured torus.
The present work considers representations of the free group on two letters in \(\text{PSL}(2,{\mathbb{C}})\). These are determined by complex solutions of the Markoff equation. Trees of the real solutions lead to Fuchsian groups discrete subgroups of \(\text{PSL}(2,{\mathbb{R}})\) as discussed by A. L. Schmidt [J. Reine Angew. Math. 286/287, 341-368 (1976; Zbl 0332.10015)]. The author seeks to understand the set of solutions which give quasi-Fuchsian groups discrete subgroups of \(\text{PSL}(2,{\mathbb{C}})\) which are geometrically finite and without accidental parabolics.
The main result in this interesting, well-written paper is the characterization of the trees of solutions which give quasi-Fuchsian groups. In brief (and glossing over certain non-trivial technicalities), these are the solutions in the “connected component” of the real solutions which (avoid groups with accidental parabolics and) satisfy a certain growth condition in their tree.