On the horoboundary and the geometry of rays of negatively curved manifolds
The authors study the Gromov boundary of quotients \(X/\Gamma\) of a Hadamard space \(X\) by an infinite group \(\Gamma\) of isometries. The authors describe some hyperbolic surfaces with infinite fundamental group whose Gromov boundaries exemplify some of the pathologies that can arise. These include divergent rays with equal Busemann functions, points on the Gromov boundary not corresponding to Busemann functions, and the discontinuity of Busemann functions as functions of the initial point and direction of a ray. For a certain class of such quotients \(X/\Gamma\), known as geometrically finite, the authors prove that the Gromov compactification has only conical singularities.
scientific article; zbMATH DE number 6109303