On asymptotic densities and generic properties in finitely generated groups

Abstract Recall that asymptotic density is a method to compute densities and/or probabilities within infinite finitely generated groups. If is a group property, the asymptotic density determines the measure of the set of elements which satisfy . Is this asymptotic density equal to 1, we say that the property is generic in G. is called an asymptotic visible property, if the corresponding asymptotic density is strictly between 0 and 1. If the asymptotic density is 0, then is called negligible. We call a group property suitable if it is preserved under isomorphisms and its asymptotic density exists and is independent of finite generating systems. In this paper we prove that there is an interesting connection between the strong generic free group property of a group G and its subgroups of finite index.