Abstract
In this paper, we prove the relative connectedness of graphs which satisfy a polynomial volume growth and a Poincare-type inequality on balls. By relative connectedness, we mean that every two vertices at distance R from a vertex x can be joined by a path within an annulus A(x, α -1 R, αR). We apply this result first to control the behavior of harmonic functions outside a ball and then, in the case of Cayley graph of groups having polynomial volume growth, to obtain a Poincare-type inequality on the annuli.
Highlights
Let r be an infinite undirected connected graph and note that we call r both the graph and its set of vertices when there is no ambiguity
We suppose that F has polynomial volume growth of exponent D: C-1RD #B(x, R) CRD.. (1.1)
Our aim is first to prove the relative connectedness of the spheres of r (Proposition 2.1) when it satisfies (1.1) and a D~Poincar~-type inequality on balls: there is a constant C(D) such that for any function u on r
Summary
Let r be an infinite undirected connected graph and note that we call r both the graph and its set of vertices when there is no ambiguity. We extend to annuli (Theorem 3.1) the elliptic Harnack inequality on balls obtained by Delmotte [2] under (1.2) and the doubling of the volume (implied by (1.1)). From this inequality, we deduce a control on the behavior of harmonic functions outside a finite set (Theorem 3.2), by comparison with the behavior of the Green function. When r is the Cayley graph of a group having polynomial volume growth, we deduce Poincare-type inequalities on annuli (Theorem 3.3)
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