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It is shown that a connected amenable group G has polynomial growth if, and only if, given any open subsemigroup S of G and a compact set K in G there exists an s in S such that Ks C S. Introduction. Let G be a connected topological group and C(G) the space of all bounded continuous real-valued functions on G. For each f in C(G) and a in G define lfJl = sup{ f(x)l: x E G} and laf(t) = f(a-1t) for all t in G. A function f in C(G) is left uniformly continuouis if whenever (si) is a net in G and converges to some s in G, then 1, f 1,f tends to zero. We say that G is amenable (more accurately, left amenable) if there exists an m in LUC(G)* the conjugate space of LUC(G) (the space of left uniformly continuous functions), such that m > 0, m = 1 and m(laf) = m(f) for all f in LUC(G) and a in G. The reader is referred to Greenleaf [3] for many interesting properties of amenable groups. Another interesting class of groups are those with polynomial growth, whose definition follows. Let IA G denote the left Haar measure of a measuirable subset A of G. A locally compact group G is said to have polynomial growth if for each compact nbd U of the identity e in G. there exists a polynomial p such that |UnJC < p(n) for all nr 1,2,3. The notion of Archimedean property for groups was introduced in [2]. A grouip G is said to have Archimedean property if whenever S is an open generating subsemigroup and K a compact subset of G there exists an s in S such that sK C S. It is proved in [2] that any connected group with polynomial growth has Archimedeail property. Some remarkably diverse properties of groups with polynomial growth can be found in [1, 4, 7]. It is well known that groups with polynomial growth are amenable. However, the converse fails. The group of affine transformations on the real line is a connected solvable group without polynomial growth. Jenkins [6] has given a necessary and sufficient condition for a connected amenable group to have polynomial growth. In this paper we give another necessary and suifficient conditioil in Theorem 2. As a corollary we note that every bounded continuious fiunction on a group with polynomial growth which is left uiniformly continuouis on a dense opeii subsemigroup is left uniformly continuouis on G. First we prove the following PROPOSITION 1. Let G be a connected locally compact group and S an open subsemigroup of G. Then the following are equivalent. Received by the c(litors October 31, 1983 and, in revise(l formii, Janiary 4, 1984. 1980 Mathematscs Subject Classification. Primiiary 22E15, 22A(5; Secondary 20M(5.

A necessary and sufficient condition for a connected amenable group to have polynomial growth