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

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 VA(Y )) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r 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 ∈ M(Y ) there is an r0(h) such that if r ≥ r0(h) and A is a primitive 4r th root of unity then h acts projectively nontrivially on VA(Y ). Jones’ [9] original representation ρn of the braid groups Bn , sometimes called the generic q–analog–SU(2)–representation, is not known to be faithful. However, we show that any braid h 6= id ∈ Bn admits a cabling c = c1, . . . , cn so that ρN (c(h)) 6= id, N = c1 + . . .+ cn . AMS Classification numbers Primary: 57R56, 57M27 Secondary: 14N35, 22E46, 53D45