  We construct certain orbifold compactifications of the moduli stack of
pointed stable curves over $\mathbb C$ and study their fundamental groups by
means of their quantum representations. This enables to construct interesting
K\"ahler groups and to settle most of the candidates for a counter-example to
the Shafarevich conjecture on holomorphic convexity proposed in 1998 by
Bogomolov and Katzarkov, using TQFT representations of the mapping class
groups.
