  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 $\pi_1^{\rm reg}(X,D)$ admits a normal abelian
subgroup of finite index and rank at most $2c$. We prove this conjecture in the
cases $0 \leq c \leq 3$, building on the almost abelianity of the fundamental
groups of klt Calabi-Yau pairs of dimension $\leq 3$. 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.
