On the number of ends of rank one locally symmetric spaces

Stover, Matthew¹

¹ Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA

Geometry & topology, Tome 17 (2013) no. 2, pp. 905-924.

Résumé

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 $n\ge 30$.

Détail

DOI : 10.2140/gt.2013.17.905

Classification : 11F06, 20H10, 22E40
Keywords: locally symmetric spaces, arithmetic lattices, rank one geometry, cusps

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