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

2006/01/06 by Pallavi Dani, Dani, Pallavi
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT

paper · pdf · doi:10.48550/arxiv.math/0601126

26 pages

arxiv created 2006/01/06 · arxiv updated 2009/12/01

Abstract

Let G be a finitely generated group with a given word metric. The asymptotic density of elements in G that have a particular property P is defined to be the limit, as r goes to infinity, 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).

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