Warped product spaces: Gromov hyperbolicity and identification of the visual boundary

Abstract

In this paper, we consider warped product spaces $X{\times }_{\phi }Y$, where $X$ is a complete geodesic Gromov hyperbolic space, $Y$ is a compact geodesic metric space, and the warping function $\phi$ satisfies suitable exponential growth conditions. We prove that the warped product is Gromov hyperbolic and derive an explicit estimate for its hyperbolicity constant. We further establish a homeomorphism between its Gromov boundary and ${\partial }_{G}X\times Y$, with an explicit comparison formula for the visual metric on the Gromov boundary of $X{\times }_{\phi }Y$ in terms of the visual metric $d_{\epsilon ,X}$ on ${\partial }_{G}X$ and $d_Y$ for suitable $\epsilon >0$.

27 pages, 2 figures