Approximating Graphs with Polynomial Growth

Let X be an in®nite, locally ®nite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an in®nite sequence of ®nite graphs satisfying growth conditions which are closely related to growth properties of the in®nite graph X. 1991 Mathematics Subject Classi®cation. Primary 05C25, Secondary 20F18 1. Introduction and statement of main result. We think of a graph X as a set of vertices, equipped with a symmetric, non re ̄exive neighbourhood relation E ˆ E…X† X X, the edge set. Graphs considered in this paper are assumed to be connected and to have bounded vertex degrees. The natural distance between two vertices x; y 2 X (the minimal number of edges on a path from x to y) is denoted by d…x; y†. An automorphism of X is a selfisometry of X with respect to this metric. The automorphism group of X is denoted by Aut…X†. The graph is called transitive if Aut…X† acts transitively on X, it is almost transitive if Aut…X† acts on X with ®nitely many orbits. For a vertex v, we write BX…v; n† for the subgraph induced by all x 2 X with d…x; v† n. The growth function of X with respect to v is fX…v; n† ˆ BX…v; n† …n ˆ 0; 1; 2; . . .†: If X is transitive, fX…n† ˆ fX…v; n† is independent of v. We say that X has polynomial growth if there are constants c, d such that fX…v; n† c n; for all n and for every vertex v. The results for ®nitely generated groups (Gromov [5]), transitive graphs (Tro®mov [10]) and locally compact groups (Losert [8]) imply that an almost transitive graph with polynomial growth is very similar to a Cayley graph of a ®nitely generated nilpotent group; see x2 below. In particular, for such a graph there are constants 0 < c1 c2 <1 and a nonnegative integer d such that for every vertex v we have c1 n fX…v; n† c2 n; for all n: …1† We say that a graph X (®nite or in®nite, not necessarily almost transitive) has the doubling property if there is a number (the doubling constant) A 1 such that Glasgow Math. J. 42 (2000) 1±8. # Glasgow Mathematical Journal Trust 2000. Printed in the United Kingdom