Regularities on the Cayley Graphs of Groups of Linear Growth

Let G be a finitely generated group and E 5 E 1 < E 2 1 a finite generating system . Define the E -length l E ( g ) of g P G as the minimum length of a representation of g as a product of elements in E , and define f E ( n ) as the number of elements in G with E -length equal to n . We will say that a finitely generated group has polynomial growth if there exists an integer k such that f E ( n ) < n k (this definition is easily seen to be independent of E ) . A deep result of Gromov says that a group of polynomial growth has a nilpotent subgroup of finite index ; or , equivalently , that it is a finite extension of a nilpotent group . Gromov’s proof is geometric and non-elementary , and it is an open problem to give a proof using only group-theoretic or combinatorial arguments . There are some results in this direction for the case of linear growth by Wilkie and Van Den Dries (see [4]) and by Justin (see [2]) , who use a combinatorial argument . Here we take a dif ferent point of view , and exhibit a regularity property on the Cayley graph of G that simplifies the proof for the linear case and seems likely to be extended to the cases of growth of higher degree .