In Riemannian geometry the non-negativity of the Ricci curvature $R$ of a manifold $X$ has strong implications on the size of the fundamental group $\pi_1(X)$:

- If $R>0$, then $\pi_1(X)$ is finite.
- If $R=0$, it is known that $\pi_1(X)$ is almost abelian, i.e., it contains an abelian subgroup of finite index. Also, $\pi_1(X)$ has polynomial growth.

In the case $X$ is a smooth complex projective variety, the positivity of Ricci curvature is related to ampleness properties of of $-K_X$, so it would be interesting to see whether analogous results of the above hold in algebraic geometry, with Ricci curvature replaced by the Kodaira dimension $\kappa(X)=\sup_n\dim\phi_{nK}(X)$. So my question is:

What implications do non-positive Kodaira dimension have for the fundamental group of $X$? In particular, does some versions of the above results hold with Ricci curvature replaced by Kodaira dimension?

For example, if $X$ is a smooth projective variety with $\kappa(X)=0$, is $\pi_1(X)$ almost abelian?

One could also ask for refined versions of the above statements. For example, when $X$ is Fano it is well-known that $\pi_1(X)=0$. Does the same condition hold for all $X$ with big $-K_X$?