Centrally extended mapping class groups from quantum Teichmuller theory

Centrally extended mapping class groups from quantum Teichmuller theory? Louis Funar Rinat M. Kashaev Institut Fourier BP 74, UMR 5582 Section de Mathematiques, Case Postale 64 University of Grenoble I Universite de Geneve 38402 Saint-Martin-d'Heres cedex, France 2-4, rue du Lievre, 1211 Geneve 4, Suisse e-mail: e-mail: March 22, 2010 Abstract The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmuller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in [9] for the Thompson groups. 2000 MSC Classification: 57M07, 20F36, 20F38, 57N05. Keywords: Mapping class group, Ptolemy groupoid, quantization, Teichmuller space, Meyer class, Euler class. Introduction The quantum theory of Teichmuller spaces of punctured surfaces of finite type, originally constructed in [4, 15] and subsequently generalized to higher rank Lie groups and cluster algebras in [7, 8], leads to one parameter families of projective unitary representations of Ptolemy modular groupoids associated to ideal triangulations of punctured surfaces. We will call such representations (quantum) dilogarithmic representations, since the main ingredient in the theory is the non-compact quantum dilogarithm function first introduced in the context of quantum integrable systems by L.

• mapping class


• group


• self-conjugate operators


• zz zz


• bb bb


• central extension


• cc cc


• compact quantum


• quantum teichmuller