On the asymptotics of quantum SU(2) representations of mapping class groups

Michael Freedman, Vyacheslav Krushkal

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We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan's property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL(2, ℤ) on L 2(ℝ). As a consequence we obtain a curious analytic fact about the Fourier transform on ℝ which may not have been previously observed.

Original languageEnglish
Pages (from-to)293-304
Number of pages12
JournalForum Mathematicum
Issue number2
StatePublished - 21 Mar 2006
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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