Homological action of the modular group on some cubic moduli spaces
Abstract
We describe the action of the automorphism group of the complex cubic x2+y2+z2-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 x2+y2+z2-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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.