Quantum SU(2) faithfully detects mapping class groups modulo center

Abstract

The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r-th root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of M(Y). (Note that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus = 2.) Specifically, for a non-central h in M(Y) there is an r_0(h) such that if r>= r_0(h) and A is a primitive 4r-th root of unity then h acts projectively nontrivially on V_A(Y). Jones' [J] original representation rho_n of the braid groups B_n, sometimes called the generic q-analog-SU(2)-representation, is not known to be faithful. However, we show that any braid h not= id in B_n admits a cabling c = c_1,...,c_n so that rho_N (c(h)) not= id, N=c_1 + ... + c_n.