Quantum traces for representations of surface groups in $\mathrm{SL}_2(\mathbb{C})$

Abstract

We relate two different quantizations of the character variety consisting of all representations of surface groups in ${SL}_2$. One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions.