Cusp and $b_1$ growth for ball quotients and maps onto $\mathbb{Z}$ with finitely generated kernel

TL;DR

This paper investigates the growth of first Betti number and cusps in towers of finite covers of ball quotients, providing explicit examples and showing the existence of lattices with homomorphisms onto al with finitely generated kernels.

Contribution

It introduces new examples of lattices in al(2,1) with homomorphisms onto al and analyzes growth rates of topological invariants in ball quotient towers.

Findings

Existence of lattices in al(2,1) with finitely generated kernels in homomorphisms onto al.

Growth rates of Betti numbers and cusps are largely independent in tower sequences.

Cocompact arithmetic lattices of simplest type contain finite index subgroups with these properties.

Abstract

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.