Celebratio Mathematica — Neumann — Complete Bibliography

Complete Bibliography

Works connected to William Mark Goldman

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W. M. Goldman and W. D. Neumann: “Homological action of the modular group on some cubic moduli spaces,” Math. Res. Lett. 12 : 4 (2005), pp. 575–591. MR 2155233 Zbl 1087.57001 ArXiv math/0402039 article

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 \( \mathrm{SL}(2,\mathbb{C}) \) character variety given by fixing the trace of the peripheral element (so-called “relative character varieties”). This mapping class group is isomorphic to \( \mathrm{PGL}(2,\mathbb{Z}) \). We also describe the corresponding mapping class group action for the four-holed sphere and its relative \( \mathrm{SL}(2,\mathbb{C}) \) 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 \( \mathrm{PGL}(2,\mathbb{Z})_{(2)} \) still acts on these cubics and is the full automorphism group when \( P \), \( Q \), \( R \) are distinct.