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I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill lots of details when I try to proceed. I would like to ask wether there are some good reference in this area, no matter in differential geometrical manner, topological manner, or algebraic manner. I posted it on stackexchange, but someone advice me to post it here. Many thanks.

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This seems a good place to start:

A survey of quantum Teichmüller space and Kashaev algebra.

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