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Kanai proved that quasi-isometries between Riemannian manifolds with bounded geometry preserve many global properties, including the existence of Green's function, i.e., non-parabolicity. However, Kanai's hypotheses are too restrictive. Herein we prove the stability of p-parabolicity (with 1 ) by quasi-isometries between Riemannian manifolds under weaker assumptions. Also, we obtain some results on the p-parabolicity of graphs and trees; in particular, we characterize p-parabolicity for a large class of trees.
bounded local geometry; p-parabolicity; parabolicity; quasi-isometries