The asymptotic density of finite-order elements in virtually nilpotent groups

Let Γ be a finitely generated group with a given word metric. The asymptotic density of elements in Γ that have a particular property P is the limit, as r→∞, of the proportion of elements in the ball of radius r which have the property P. We obtain a formula to compute the asymptotic density of finite-order elements in any virtually nilpotent group. Further, we show that the spectrum of numbers that occur as such asymptotic densities consists of exactly the rational numbers in [0,1).