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.

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Published 1 January 2005