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Mar 29, 2022 at 19:20 comment added Sam Hopkins Sorry I can't point to something more specific, but possibly the large literature on cluster algebras from surfaces would contain some of the answers to your questions (in particular I believe the mapping class group action is roughly speaking the same as the "cluster automorphism" group action).
Mar 29, 2022 at 18:48 history asked giulio bullsaver CC BY-SA 4.0