Images of Quantum Representations of Mapping Class Groups and Dupont–guichardet–wigner Quasi-Homomorphisms

Abstract

International audienceWe prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann. 294 (1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France 133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations

Quotients of the mapping class group by power subgroups

“…In [17], Pitsch and the second author used quantum topological techniques to prove that Mod g / Mod g [p] is not commensurable to a higher rank lattice whenever g 4 and p 2g − 1.…”

“…In , Pitsch and the second author used quantum topological techniques to prove that is not commensurable to a higher rank lattice whenever and . The first purpose of this note is to give the following uniform version of this result, in the case of once‐punctured surfaces: Theorem Let , where if , and if .…”

Profinite Completions of Burnside-Type Quotients of Surface Groups

“…Finally notice that G g,p,(p−3) contains G g−1,p as a subgroup. In particular, for g ≥ 4 each noncompact factor has rank at least 2, by [12]. We can follow the proof of this result in [12] for i = 0 to obtain the result for g = 3 as well.…”

“…Since Γ 1 g is perfect when g ≥ 3 and of order 10 for g = 2, it follows that ρ p,(i) ( Γ 1 g ) is contained within the special unitary group SU g,p,ζ,(i) , if (g, p) = (2, 5), as in [5,11,12].…”

“…When p is prime p ≥ 5 and g ≥ 2, (g, p) = 5, then ρ p,ζp takes values in the special unitary group SU g,p,ζp (see [5,11,12]). It is known that SU g,p,ζ (O p ) is an irreducible lattice in a semi-simple algebraic group G g,p obtained by the so-called restriction of scalars construction from the totally real cyclotomic field Q(ζ p + ζ p ) to Q.…”