Volumes of Discrete Groups and Topological Complexity of Homology Spheres

Can one ascribe a “size” to a discrete group? By that we mean a real-valued invariant, μ(Γ) satisfying the following condition: for any subgroup ∆ < Γ of finite index, μ(∆) = (Γ : ∆)μ(Γ). The importance of such an invariant for the structure of Γ is clear from the following observation: if μ(Γ) > 0 and ∆1,∆2 are two isomorphic subgroups of Γ of finite index, then (Γ : ∆1) = (Γ : ∆2). Some rational-valued invariants satisfying the axiom above, have been studied by Wall [17], Bass [1], Chiswell [7], Brown [4], Stallings [16] etc. and called (generalized) Euler characteristics. These are defined for a FP group Γ by χ(Γ) = ∑ (−1)rkPi where P → Q is a finite f.g. projective resolution of Q and rk : K0(QΓ) → Z is a homomorphism and extended to all VFP groups in the usual way [17]. On the other hand, if Γ runs through fundamental groups of closed three-manifolds, there are two known invariants, (usually called covering invariants by obvious reason). One is Gromov’s simplicial volume, which is positive for any hyperbolic manifold. The other is Wang and Wu’s invariant [18], defined for any plumbing of Seifert fibrations. The approach presented in this paper, takes a new standpoint in dealing the above problem, that is, a “measure-theoretic” one. Namely, we start with “semimultiplicative invariants”, analogous to (outer) measure in measure theory and then produce a truly multiplicative invariants by going to lower limit over the directed set of finite index subgroups. In course of application, we address two fundamental and well-known problems of Gromov and Lyndon: