Vol. 259, No. 1, 2012

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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

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.

Gromov boundary, horofunction, compactification, geometrically finite manifold, asymptotic ray
Mathematical Subject Classification 2010
Primary: 53C20, 53C22, 57S30
Secondary: 53A35, 53C23
Received: 11 January 2011
Accepted: 6 July 2012
Published: 31 August 2012
Françoise Dal’bo
Institut de recherche mathématique de Rennes
Université de Rennes 1
Campus de Beaulieu
35042 Rennes
Marc Peigné
Faculté des Sciences et Techniques
Université François Rabelais
Parc de Grandmont
37200 Tours
Andrea Sambusetti
Istituto di Matematica “G. Castelnuovo”
Università “La Sapienza” di Roma
P.le Aldo Moro 5
00185 Roma