On the horoboundary and the geometry of rays of negatively curved manifolds

Françoise Dal’bo, Marc Peigné and Andrea Sambusetti

Vol. 259 (2012), No. 1, 55–100

DOI: 10.2140/pjm.2012.259.55

Abstract

We study the Gromov compactification of quotients X∕G of a Hadamard space X by a discrete group of isometries G, pointing out the main differences with the simply connected case. We prove a criterion for the Busemann equivalence of rays on these quotients and show that the “visual” description of the Gromov boundary breaks down, producing examples for the main pathologies that may occur in the nonsimply connected case, such as: divergent rays having the same Busemann functions, points on the Gromov boundary that are not Busemann functions of any ray, and discontinuity of the Busemann functions with respect to the initial conditions. Finally, for geometrically finite quotients X∕G, we recover a simple description of the Gromov boundary, and prove that in this case the compactification is a singular manifold with boundary, with a finite number of conical singularities.

Keywords

Gromov boundary, horofunction, compactification, geometrically finite manifold, asymptotic ray

Mathematical Subject Classification 2010

Primary: 53C20, 53C22, 57S30

Secondary: 53A35, 53C23

Milestones

Received: 11 January 2011

Accepted: 6 July 2012

Published: 31 August 2012