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# Mathematical Research Letters

## Volume 12 (2005)

### Number 4

### Homological action of the modular group on some cubic moduli spaces

Pages: 575 – 591

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n4.a11

#### Authors

#### Abstract

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.