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Based on the lectures given by the author at the School on braids and low dimensional topology “Winter Braids VI”, University of Lille I, 22-25 February 2016, we review the combinatorics underlying the Teichmüller TQFT, a new type of three-dimensional TQFT with corners where the vector spaces associated with surfaces are infinite dimensional. The geometrical ingredients and the semi-classical behaviour suggest that this theory is related with hyperbolic geometry in dimension three.

Combinatorics of the Teichmüller TQFT