Introduction

Let G be a finitely generated (infinite) virtually free group and denote by mo the least common multiple of the orders of the finite subgroups in G. The purpose of this note is to survey some recently obtained results centring around the arithmetic function be : N → N,

bG(λ)= number of free subgroups of index λmG in G, which indicate a quite intimate relationship between combinatorial and structural properties of G. Proofs and more details will be given elsewhere. Our main results include an asymptotic formula for bG(λ) in terms of a presentation of G as a fundamental group of a finite graph of finite groups in the sense of Bass and Serre and an explicit estimate on the growth behaviour of the difference function bG(λ+1) − bG(λ). Furthermore we consider the question as to what kind of information on the structure of G is contained in the number of free subgroups of finite index and as to what extent this combinatorial information determines the group G.

I would like to thank Robert Bieri and Heinrich Niederhausen for stimulating discussions during the preparation of this work.

Connection with Stall ings' structure theorem

Our approach to be is to relate it to another arithmetic function ao: N → Q which is certainly easier to compute. By a torsion-free G-action on a set X we mean a Gaction on X which is free when restricted to finite subgroups. For a finite set X to admit a torsion-free G-action it is necessary and sufficient that |X| be divisible by me. Now let μ be a positive integer.