On the TQFT representations of the mapping class groups

Abstract

We prove that the image of the mapping class group by the representations arising in the SU (2)-TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r = 2, 3, 4, 6 (and r = 10 for g = 2). In particular it follows that the quotient groups M g /N (t r ) by the normalizer of the r-th power of a Dehn twist t are infinite if g ≥ 3 and r = 2, 3, 4, 6, 8, 12.

Two questions on mapping class groups

“…This is the SO(3) quantum representation in level 5 (see e.g. [13]), whose dimension d g is given by the Verlinde formula: Actually M 2 has a finite index subgroup which surjects onto a free non-abelian group and hence it does not have property F A 1 . The situation is subtler for g ≥ 3, and it seems unknown whether finite index subgroups of M g have property F A 1 .…”

“…The second inequality comes from the existence of quantum representations of M g with infinite image ( [13]). …”

Problems on invariants of knots and 3–manifolds

“…In higher genus, it is known [132,274] that the image of M g (g ≥ 2) is infinite in general. Remark The Nielsen-Thurston classification says that any mapping class of a surface is either finite order, reducible, or pseudo-Anosov (see, e.g.…”

Integral topological quantum field theory for a one-holed torus

“…Thus the h-adic expansion of the TQFT-representation ρ p is complete in the sense that if a mapping class is detected by ρ p then it is also detected by ρ p,N for all big enough N . The image of ρ p is usually infinite [Funar 1999;Masbaum 1999]. In this case, the images of ρ p,N are finite groups whose size must increase as N → ∞.…”