On the number of ends of rank one locally symmetric spaces

Abstract

Let $Y$ be a noncompact rank one locally symmetric space of finite volume. Then $Y$ has a finite number $e\left(Y\right)>0$ of topological ends. In this paper, we show that for any $n\in \mathbb{N}$, the $Y$ with $e\left(Y\right)\le n$ that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant $c_n$ such that $n$–cusped arithmetic orbifolds do not exist in dimension greater than $c_n$. We make this explicit for one-cusped arithmetic hyperbolic $n$–orbifolds and prove that none exist for $n\ge 30$.