On a sequence of singular ball quotient surfaces on the line $K^2=9\chi -18$

Starting from computer experiments with the fundamental group of the Cartwright–Steger surface, we construct an infinite tower $(X_n{)}_{n\ge 1}$ of normal projective surfaces obtained by successive $\mathbb{Z}\mathrm{/}3$-Galois covers $X_n\to X_{n-1}$. For $n>1$, their minimal resolutions ${\tilde{X}}_n$ lie on the line $K^2=9\chi -18$ (equivalently $c_1^2=3c_2-72$), which is parallel to the Bogomolov–Miyaoka–Yau line $K^2=9\chi$ of ball quotients. We compute the fundamental groups for the first cases, showing that ${\pi }_1({\tilde{X}}_n)=1$ for $n=1,\dots ,5$. Motivated by the geometry of the construction, we conjecture that all ${\tilde{X}}_n$ are simply connected.