There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...

Abstract. We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph a.s. admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, a.s. the infinite clusters of p-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when p is sufficiently close to 1. Many conjectures and questions are presented. §1. Introduction. The question of whether various potential-theoretic properties of graphs and manifolds are preserved under perturbations or approximations has been studied for more than a

. We survey the field of uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free or wired boundary conditions. Among other results, Pemantle (1991) proved that in Z d , the free and wired spanning forests coincide and that they give a single tree iff d 4. The theory has developed considerably since then and found further connections to random walks, potential theory, harmonic Dirichlet functions, invariant percolation and amenability. A crucial new tool is an algorithm invented by Wilson (1996) to generate random spanning trees. Uniform spanning forests also yield insights into loop-erased walks and harmonic measure from infinity. x1. Introduction. We begin with a brief history of this fertile and fascinating field. In subsequent sections, we will more carefully define and develop most of what we recount here. More details for much of the material surveyed here ...

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs.

Abstract. Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L p-cohomology space of some groups that have one end. We also make a connection between the first L p-cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L p-cohomology space of a nonelementary hyperbolic group does not vanish. 1.

Abstract. Let G be a finitely generated infinite group and let p> 1. In this paper we make a connection between the first L p-cohomology space of G and p-harmonic functions on G. We also describe the elements in the first L p-cohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups. 1.

Abstract. Let p be a real number greater than one and let G be a finitely generated, infinite group. In this paper we introduce the p-harmonic boundary of G. We then characterize the vanishing of the first reduced ℓ p-cohomology of G in terms of the cardinality of this boundary. Some properties of p-harmonic boundaries that are preserved under rough isometries are also given. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on G, the p-harmonic boundary of G and the first reduced ℓ p-cohomology of G. 1.