On the number of ends of rank one locally symmetric spaces

Stover, Matthew

doi:10.2140/gt.2013.17.905

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On the number of ends of rank one locally symmetric spaces

Matthew Stover

Geometry & Topology 17 (2013) 905–924

DOI: 10.2140/gt.2013.17.905

Abstract

Let $Y$ be a noncompact rank one locally symmetric space of finite volume. Then $Y$ has a finite number $e (Y) > 0$ of topological ends. In this paper, we show that for any $n \in ℕ$ , the $Y$ with $e (Y) \leq n$ 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 $n \geq 30$ .

Keywords

locally symmetric spaces, arithmetic lattices, rank one geometry, cusps

Mathematical Subject Classification 2010

Primary: 11F06, 20H10, 22E40

References

Publication

Received: 19 September 2012

Accepted: 30 January 2013

Published: 30 April 2013

Proposed: Benson Farb

Seconded: Ronald Stern, John Lott

Authors

Matthew Stover
Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109-1043 USA
http://www.math.lsa.umich.edu/~stoverm

Volume 17, issue 2 (2013)