THE NORMALIZED CYCLOMATIC QUOTIENT ASSOCIATED WITH PRESENTATIONS OF FINITELY GENERATED GROUPS

Given a presentation of an n-generated group, we define the normalized cyclomatic quotient (NCQ) of it, which gives a number between 1-n and 1. It is computed through an investigation of the asymptotic behavior of a kind of an ''average rank'', or more precisely the quotient of t he rank of the fundamental group of a finite subgraph of the correspon ding Cayley graph by the size of the subgraph. In many ways (but not a lways) the NCQ behaves similarly to the behavior of the spectral radiu s of a symmetric random walk on the graph. In particular, it character izes amenable groups. For some types of groups, like finite, amenable or free groups, its value equals that of the Euler characteristic of t he group. We give bounds for the value of the NCQ for factor groups an d subgroups, and formulas for its value on direct and free products. S ome other asymptotic invariants are also discussed.