A conjecture of Ghys
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Let M be a compact manifold. Ghys conjectured that all finite subgroups of the diffeomorphism group Diff(M)
have an abelian normal subgroup whose index is bounded by a constant depending on M.
This turned out to be false, so he updated the conjecture, replaced the word "abelian" with "nilpotent".
In the talk I'll discuss both versions of the conjecture, as well as related questions in algebraic geometry.
The talk is based on a joint work with B. Csikós and L. Pyber.