We study the geometry of infinitely presented groups satisfying the small cancellation condition $C^{\prime }(1/8)$, and introduce a standard decomposition (called the criss-cross decomposition) for the elements of such groups. Our method yields a direct construction of a linearly independent set of power continuum in the kernel of the comparison map between the bounded and the usual group cohomology in degree 2, without the use of free subgroups and extensions.

We 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.