The $p$-harmonic boundary for quasi-isometric graphs and manifolds

Let $p$ be a real number greater number greater than one. Suppose that a graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$ with certain properties. Under these conditions we will show that the $p$-harmonic boundary of $G$ is homeomorphic to the $p$-harmonic boundary of $M$. We will also prove that there is a bijection between the $p$-harmonic functions on $G$ and the $p$-harmonic functions on $M$.