Homological action of the modular group on some cubic moduli spaces

We describe the action of the automorphism group of the complex cubic x^2+y^2+z^2-xyz-2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2,C) character variety given by fixing the trace of the peripheral element (so-called "relative character varieties"). This mapping class group is isomorphic to PGL(2,Z). We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2,C) character varieties, which are fibers of deformations x^2+y^2+z^2-xyz-2-Px-Qy-Rz of the above cubic. The 2-congruence subgroup PGL(2,Z)_(2) still acts on these cubics and is the full automorphism group when P,Q,R are distinct.