Homological action of the modular group on some cubic moduli spaces

Abstract

We describe the action of the automorphism group of the complex cubic x + y + x − 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 (socalled “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 + y + z − 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.