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The goal of this note is to introduce an interesting question proposed by D.Rydh on an analytic version of the local structure of Artin stacks saying thatnear points with linearly reductive stabilizers, Artin stacks are\'etale-locally quotient stacks. We give some supporting evidence by verifyingit on two fundamental classes of classical analytic moduli spaces: theTeichm\"uller moduli space and the Riemann moduli space of integrable complexstructures whose analytic stack versions have been constructed by a recent workof L. Meersseman.

Local structure of the Teichm\"uller and the Riemann moduli stacks