Restrictions on $\pi_1(X)$ of geometric origin (Kähler groups as example)

Question

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?

Answer 1

Some properties of Kähler groups are retained by other classes of closely related groups, such as fundamental groups of smooth, quasi-projective varieties, or fundamental groups of Sasakian manifolds.

For instance, if $M$ is a smooth, quasi-projective variety, then the first characteristic variety $V^1_1(M)$, i.e., the set of characters $\rho\colon \pi_1(M)\to \mathbb{C}^*$ for which $H^1(M, \mathbb{C}_{\rho})\ne 0$, is a finite union of torsion-translated subtori of the character group $\operatorname{Hom}(\pi_1(M), \mathbb{C}^*)$.

As another example, if $M$ is a compact Sasakian manifold, then the same conclusion holds for the irreducible components of $V^1_1(M)$ passing through the identity. Moreover, if $\dim(M)\ge 5$, then $\pi_1(M)$ is $1$-formal, i.e., its Malcev Lie algebra is quadratic.