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Abstract: In the 19th century, Vogt (and later Fricke and Klein) showed that the GIT quotient S of
SL(2) x SL(2,C) by Inn(SL(2,C) ) is an affine space C^3. Fricke and Klein were motivated by
uniformizing Riemann surfaces by hyperbolic non-Euclidean geometry. Viewing SL(2) x SL(2) as Hom(F2, SL(2))
(where F2 is the two-generator free group) leads to an algebraic action of Out(F2) (which is isomorphic
to the modular group GL(2,Z)) on X. This action has interesting dynamical properties. In particular this action
preserves the trace of the commutator of the free generators of F2, which is the cubic polynomial k = x^2 + y^2 + z^2 - xyz - 2
For simplicity we consider only the set of R-points, although the C-points are extraordinarily rich and complicated.
The topology bifurcates at the level sets of the critical values of k, which are +2 and -2 --- these are already
interesting and classically studied: integer points on the level set k=-2 are the Markoff triples,
and the level set k=-2 is the Cayley cubic characterized by having 4 nodes and a rational parametrization of its
affine patch. The dynamics bifurcates at the level k =18, which turns out to be an affine patch on the famous Clebsch diagonal surface.
We relate the geometry and dynamics of these affine cubics to their classical projective geometry,
also developed in the 19th century by Cayley, Salmon, Schlafli, Sylvester, and Cremona.