A Remark About the Spectral Radius

Abstract

We show that for any finitely generated non-amenable group and any e > 0, there exists some finite symmetric generating set with spectral radius less than e. We give applications to percolation theory and the theory of operator spaces. 1. The spectral radius Let Γ be a finitely generated group and let S ⊂ Γ be a finite symmetric generating set. Here, a set is called symmetric if S−1 = S, i.e. g ∈ S implies g−1 ∈ S. Consider the Hilbert space `Γ with orthonormal basis {δg | g ∈ Γ} and the left-regular representation λ : Γ→ U(`Γ), which is defined by the formula λ(g)δh := δgh. We will also use λ to denote the linear extension λ : C[Γ]→ B(`Γ). Here C[Γ] denotes the complex group ring and λ is a ∗-homomorphism with respect to the natural involution on C[Γ]. There is a natural trace τ : C[Γ]→ C, defined by τ( ∑ g agg) = ae. For each a = ∑ g agg ∈ C[Γ], we denote by ‖a‖ the operator norm of the operator λ(a) ∈ B(`Γ). We also set supp(a) := {g ∈ Γ | ag 6= 0}, size(a) := |supp(a)| and ‖a‖1 := ∑ g |ag|. It is a basic property of the operator norm that ‖a‖ = ‖a‖ whenever a is a hermitean element, i.e. a∗ = a. For a, b ∈ R[Γ], we write a ≤Γ b if ag ≤ bg for all g ∈ Γ. If 0 ≤Γ b ≤Γ a and a, b ∈ R[Γ] are hermitean, then ‖b‖ ≤ ‖a‖. Indeed, this follows from the spectral radius formula (1) ‖b‖ = lim n→∞ τ(b) ≤ lim n→∞ τ(a) = ‖a‖. Clearly, if 0 ≤Γ a, then ‖a‖1 = ‖a‖1 . For any symmetric subset S ⊂ Γ, we define the Markov operator m(S) := 1 |S| ∑ s∈S s. Kesten [5] showed that the group Γ is non-amenable if and only ‖m(S)‖ < 1 for some (and hence any) finite symmetric generating set of Γ. We will also use the notation ρ(S) := ‖m(S)‖ and call it the spectral radius of the random walk associated with S. The spectral radius formula ρ(S) = lim n→∞ τ(m(S)) gives the explanation for this terminology, since the right side of this equation is the exponential growth rate of the return probability of a random walk in the Cayley graph of Γ with respect to the generating set S after 2n steps, starting at the neutral element, see [11] for definitions and references. 1