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The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization
- The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization
- Kim, Hyun Kyu
- Ewha Authors
- Issue Date
- Journal Title
- ADVANCES IN MATHEMATICS
- ADVANCES IN MATHEMATICS vol. 293, pp. 529 - 588
- Quantum Teichmuller theory; Thompson group T; Ptolemy-Thompson group; Kashaev quantization; Braided Ptolemy-Thompson group; Universal Teichmuller space; Stable braid group; Infinite braid group
- ACADEMIC PRESS INC ELSEVIER SCIENCE
- SCI; SCIE; SCOPUS
- Document Type
- Quantization of universal Teichmuller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T by Z, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension (T) over cap (Kash) of T resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in H-2 (T; Z). Meanwhile, the braided Ptolemy-Thompson groups T*, T-# of Funar-Kapoudjian are extensions of T by the infinite braid group B-infinity and by abelianizing the kernel B-infinity one constructs central extensions T-ab*, T-ab(#) of T by Z, which are of topological nature. We show (T) over cap (Kash) congruent to T-ab(#) Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension (T) over cap (CF) of T resulting from the Chekhov-Fock (-Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that (T) over cap (CF) congruent to T-ab*. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations. (C) 2016 Elsevier Inc. All rights reserved.
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