#### Volume 17, issue 2 (2013)

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On the number of ends of rank one locally symmetric spaces

### Matthew Stover

Geometry & Topology 17 (2013) 905–924
##### 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 $n\in ℕ$, 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 $n\ge 30$.

##### Keywords
locally symmetric spaces, arithmetic lattices, rank one geometry, cusps
##### Mathematical Subject Classification 2010
Primary: 11F06, 20H10, 22E40