Geodesic
The pentagon equation and mapping-class groups of punctured surfaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 198-204

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In the quantum Teichmüller theory, the mapping-class groups of punctured surfaces are represented projectively based on Penner coordinates. Algebraically, the representation is based on the pentagon equation together with pair of additional relations. Two more examples of solutions of these equations are connected with matrix (or operator) generalizations of the Rogers dilogarithm. The corresponding central charges are rational. It is possible that this system of equations admits many different solutions. 
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     author = {R. M. Kashaev},
     title = {The pentagon equation and mapping-class groups of punctured surfaces},
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R. M. Kashaev. The pentagon equation and mapping-class groups of punctured surfaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 198-204. http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a4/