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(Y ) > 0 of topological ends. In this paper, we show that for any n , the Y with e(Y ) n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant cn such that n–cusped arithmetic orbifolds do not exist in dimension greater than cn. We make this explicit for one-cusped arithmetic hyperbolic n–orbifolds and prove that none exist for n 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