Parabolicity on Graphs

Abstract

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green’s function (i.e. a positive fundamental solution of the p-Laplace-Beltrami operator). Herein we study directly the p-parabolicity on graphs. We obtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1 < p < ∞.