For complex smooth algebraic surfaces of general type the Bogomolov--Miyaoka--Yau inequality \(K^2 \le 9\chi\) holds.
Surfaces on the line \(K^2 = 9\chi\) are ball quotients and have infinite fundamental group, and it is natural to ask how close one can get to this line using simply connected surfaces.

In this talk, I will explain how computer experiments with the fundamental group of a ball quotient surface, the Cartwright--Steger surface, led us to a geometric construction of an infinite tower of surfaces on the line $K^2 = 9\chi - 18,$ which is parallel and asymptotically close to the Bogomolov--Miyaoka--Yau line.

The experiments revealed a recurring pattern of index--\(3\) subgroups, suggesting the existence of successive \(\mathbb{Z}/3\)-covers.
Guided by this observation, we construct a sequence of surfaces using triple covers branched on suitable configurations of singularities. We show that the first few surfaces are simply connected and conclude with a conjecture that this holds for the entire tower.