On the number of ends of rank one locally symmetric spaces

TL;DR

This paper investigates the number of ends of rank one locally symmetric spaces, establishing finiteness results for arithmetic cases and providing explicit bounds for hyperbolic orbifolds in high dimensions.

Contribution

It proves that arithmetic rank one locally symmetric spaces with bounded ends fall into finitely many classes and explicitly bounds the dimension where n-cusped hyperbolic orbifolds cease to exist.

Findings

Finiteness of commensurability classes for spaces with bounded ends

Explicit bound c_n for the non-existence of n-cusped hyperbolic orbifolds in high dimensions

No 30-dimensional or higher 1-cusped arithmetic hyperbolic orbifolds exist

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 natural number 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.