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Images of quantum representations of mapping class groups and Dupont-Guichardet-Wigner quasi-homomorphisms. (English) Zbl 1390.57008

Let \(M_g\) be the mapping class group of an oriented surface of genus \(g\). Let \(\mathcal{V}_p\) be a Topological Quantum Field Theory (TQFT) of \(M_g\). The following data are associated to \(\mathcal{V}_p\): a projective representation \(\rho_{p,\xi}\), where \(p\) is an integer \(\geq 5\) and \(\xi\) is a suitable root of unity, a suitable linear representation \(\tilde{\rho}_{p,\xi}\) of the central extension \(\widetilde{M}_g\), and a \(\mathbb{C}\)-vector space of conformal blocks each of dimension \(N(g,p)\). Moreover, let \(\mathcal{O}_p=\mathbb{Z}[\xi_p]\) denote the ring of cyclotomic integers, for a prime \(p\geq5\), and \(p\equiv -1\) (mod 4), there is a free \(\mathcal{O}_p\)-lattice, \(S_{p,g}\), in the space of conformal blocks and a non-degenerate Hermitian \(\mathcal{O}_p\)-valued form on \(S_{p,g}\) invariant under the action of \(\widetilde{M}_g\) via \(\tilde{\rho}_{p,g}\). For suitable roots of unity, the representation \(\tilde{\rho}_{g,\xi}\) takes values in a certain semisimple real algebraic group \(\mathbb{G}_{g,p}\) defined over \(\mathbb{Q}\). The representations \(\rho_{p,g}\) and \(\tilde{\rho}_{p,g}\) define representations \(\rho_p\) and \(\tilde{\rho}_p\) over \(\mathbb{G}_{p,g}\). This paper is concerned with the image of \(\rho_{g,\xi}\) and its kernel. Let \(\widetilde{M}^u_{g}\) denote the universal central extension of \(M_g\). Let \(s_{p,g}\) denote the number of simple non-compact factors of \(\mathbb{G}_{p,g}\) and \(r_{p,g}\) denote the minimum number of relators to be added in order to obtain the quotient \(\tilde{\rho}_p(\widetilde{M}^u_g)\). The first main result is the following: Let \(g\geq 4\), and \(p\) be a prime number with \(p\equiv -1\) (mod 4). Then, one of the following holds: either \(\tilde{\rho}_p(\widetilde{M}^u_g)\) is not isomorphic to a higher rank lattice, or \(r_{p,g}\geq s_{p,g}\). Moreover, \[ s_{p,g}\geq \lceil \frac{g-3}{2(g-1)}p+\frac{3}{2} \rceil\text{ for } p\geq 2g-1, \] where \(\lceil x\rceil \) denotes the smallest integer greater than or equal to \(x\). As a corollary the authors provide examples where \(M/M[p]\) is not isomorphic to a higher rank lattice, where \(M[p]\) denotes the normal subgroup of \(M_p\) generated by the \(p\)-th powers of Dehn twists. Moreover, the authors give many instances for which \(H^2(\tilde{\rho}_p(\widetilde{M}^u_g);\mathbb{R})\geq 1\) and \(H^2(\tilde{\rho}_p(\widetilde{M}^u_g);\mathbb{R})\geq 1\) as well.

MSC:

57M50 General geometric structures on low-dimensional manifolds
55N25 Homology with local coefficients, equivariant cohomology
19C09 Central extensions and Schur multipliers
20F38 Other groups related to topology or analysis
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