There's and old and extensively studied question about characterisation of fundamental groups of smooth compact Kähler manifolds. Restrictions imposed by Kählerness are somewhat fragile, and if we drop some properties and go aside a bit, every finitely presented group can be realized as a $\pi_1$ of 1) symplectic complex threefold — no relation between $I$ and $w$, however (an old result by Gompf); 2) projective surface with mild singularities (M. Kapovich paper https://www.math.ucdavis.edu/~kapovich/EPR/tiling.pdf); 3) 6-dimensional symplectic manifold with trivial $c_1$ (A. Petrunin afair).
Let dimensions of our manifolds to be $\geq 4$ to omit purely dimensional properties of $\pi_1$. Are there some geometric structures (say, defined by collection of forms, e. g. quaternionic Kähler, hermitian almost complex, CR manifolds with additional properties — you name it) or "locally defined" constraints of different kind (Chern class relations, comparison morphisms between cohomology of natural sheaves) for which fundamental groups of smooth manifolds of that kind have non-obvious nice properties such as superrigidity in symmetric spaces of some kind?
More hand-wavy version: is there some differential-geometric description for coastline of country where manifolds with restricted fundamental group live?