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 4rth 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 ε M(Y) there is an r0(h) such that if r ≥ r0(h) and A is a primitive 4rth 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 ≠ id ε Bn admits a cabling c = c1, . . . , cn so that ρN(c(h)) ≠ id, N = c1 + . . . + cn.