On a sequence of singular ball quotient surfaces on the line $K^2=9χ-18$
Carlos Rito, Xavier Roulleau
TL;DR
The paper constructs an infinite tower of normal surfaces starting from the Cartwright--Steger surface via successive $\mathbb{Z}/3$-Galois covers, producing minimal resolutions $\widetilde{X}_n$ that lie on the line $K^2=9\chi-18$ for $n>1$. It combines computational experiments on the Cartwright--Steger fundamental group with a geometric framework of 3-divisible singularities and Galois covers to analyze invariants and fundamental groups, proving that the first five $\widetilde{X}_n$ are simply connected and conjecturing that all are. The construction yields alternating towers of surfaces $S_n$ (ball quotients) and $X_n$, connected by $(\mathbb{Z}/3)^2$-covers, and demonstrates that the limit behavior approaches the Bogomolov--Miyaoka--Yau line while remaining near simply connected territory. A supplementary variant using $\frac{1}{3}(1,1)$-type singularities leads to $W_n$ with different invariants and initial fundamental groups, illustrating the method’s flexibility. The results advance understanding of geography near the ball quotient boundary and suggest a broad class of simply connected examples in proximity to the B-M-Y line.
Abstract
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 $\mathbb Z/3$-Galois covers $X_{n}\to X_{n-1}$. For $n>1$, their minimal resolutions $\widetilde{X}_n$ lie on the line $K^2 = 9χ- 18$ (equivalently $c_1^2 = 3c_2 - 72$), which is parallel to the Bogomolov--Miyaoka--Yau line $K^2 = 9χ$ of ball quotients. We compute the fundamental groups for the first cases, showing that $π_1(\widetilde{X}_n)=1$ for $n=1,\ldots,5$. Motivated by the geometry of the construction, we conjecture that all $\widetilde{X}_n$ are simply connected.