Freedman, Michael; Krushkal, Vyacheslav On the asymptotics of quantum SU(2) representations of mapping class groups. (English) Zbl 1120.57014 Forum Math. 18, No. 2, 293-304 (2006). Mapping class groups of surfaces play an fundamental role in low-dimensional topology and are only now beginning to be systematicly understood. The present paper concerns the representation theory of mapping class groups, and in particular those arising from (2+1)-dimensional topological quantum field theory. Due to the many analogies between mapping class groups and arithmetic groups, it is natural to ask whether mapping class groups have Kazhdan’s property (T), and in particular whether the quantum \(SU(2)\)-representations have almost invariant vectors. (That property (T) fails, and that an infinite direct sum of quantum \(SU(2)\)-representations does possess almost invariant vectors, was recently announced by J. Andersen [Mapping class groups do not have Kazhdan’s property (T), math.QA:0706.2184.] The present paper gives a construction of almost invariant vectors in some special cases, in particular for the hyperelliptic mapping class groups, and for the classical modular group \(SL(2,{\mathbb Z})\), which is the mapping class group of the torus. The first main result is the existence of almost invariant vectors for the quantum \(SU(2)\)-representation for the mapping class group of \(S^2\) with \(n\) distinguished points. The authors deduce that the hyperelliptic mapping class group (the centralizer of a hyperelliptic involution) and the the spherical mapping class groups don’t have property (T). The next result is that the quantum \(SU(2)\)-representations of the mapping class group of the torus \(T^2\) converge (as the level \(k\longrightarrow \infty\)), to an irreducible component of the metaplectic representation of \(SL(2,\mathbb Z)\) on \(L^2(R)\). Contrasting the situation for the marked sphere, the quantum representations for the torus have no almost invariant vectors. Finally the authors give an application to Fourier analysis: there exists a set of real numbers having constant density, such that no set similar to it supports functions almost invariant under Fourier transform. The paper is reasonably self-contained, and gives a nice introduction to Kazhdan’s property (T) for the purposes needed here. Reviewer: William Goldman (College Park) Cited in 7 Documents MSC: 57R56 Topological quantum field theories (aspects of differential topology) 22E46 Semisimple Lie groups and their representations 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type Keywords:Quantum representations; mapping class group; Kazhdan property T; almost invariant vectors; Fell topology; braid group; metaplectic representation PDFBibTeX XMLCite \textit{M. Freedman} and \textit{V. 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