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The central extension of the Thompson group  T  that arises in the quantized Teichmüller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so‐called braided Ptolemy–Thompson group introduced and studied in Funar and Kapoudjian,  Geometry & Topology  12 (2008) 375–430. We then describe the cyclic central extensions of  T  by means of explicit presentations.

Central extensions of the Ptolemy–Thompson group and quantized Teichmüller theory