In this paper we introduce a quasi-isometric invariant class of metric spaces which we call metrically thick. We show that any metrically thick space is not (strongly) relatively hyperbolic with respect to any non-trivial collection of subsets. Further, we show that the property of being (strongly) relatively hyperbolic with thick peripheral subgroups is a quasi-isometry invariant. We show that...

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for which ∪Gn = G. Pemantle’s arguments (1991) imply that the uniform measures on spanning trees of Gn converge weakly to an Aut(G)-invariant measure μG on essential spanning forests of G. We show that...

In this paper, we give an extension of the Wendel's theorem on KPC-hypergroups. We also show that every translation invariant mapping is corresponding with a unique positive measure on the KPC-hypergroup.