Let be a group with generators and let be the
corresponding Cayley graph. For any finite subgraph in ,
let be the average vertex degree of . By
we denote the least upper bound of all the , where
runs over all finite subgraphs in .
It is well-known that is amenable if and only if . There exists one more criterion of amenabilty. Namely, is non-amenable if and only if there exists a function such that 1) the distance between and in is bounded by a constant for all and 2) each element in has at least two preimages under .
Suppose that in the above definition. This means that for any , the element is either , or equals , where is a generator or its inverse. We say that the graph has a doubling property whenever the function with the above conditions exists for .
The group can be defined by the following group presentation:
1) has a doubling property if and only if . (This is in fact true for any -generated group.)
2) There exists a family of finite subgraphs in such that . In particular, .
We conjecture that for the Cayley graph of in generators , . If this conjecture is true, then is not amenable.
Besides, we present an easy formula to find the length of a given
element in generators , .
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.