Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs

In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary dimension) the fundamental group ${\pi }_1^{\mathrm{r}\mathrm{e}\mathrm{g}}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We prove this conjecture in the cases $0\le c\le 3$, building on the almost abelianity of the fundamental groups of klt Calabi-Yau pairs of dimension $\le 3$. In the cases $c\in \{0,1,2\}$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. In dimension three, we are able to prove almost abelianity of the fundamental group of the regular locus for projective klt Calabi-Yau pairs.