We prove that if a metric space satisfies a suitable growth condition in the quasihyperbolic metric and the Gehring–Hayman theorem in the original metric, then the inner boundary of the space is homeomorphic to the Gromov boundary. Thus, the inner boundary is compact.
"Quasihyperbolic boundary condition: Compactness of the inner boundary." Illinois J. Math. 55 (3) 1221 - 1233, Fall 2011. https://doi.org/10.1215/ijm/1371474552