Multiplicative measures on free groups

Alexandre V. Borovik, Alexei G. Myasnikov, Vladimir N. Remeslennikov

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24 Scopus citations

Abstract

We introduce a family of multiplicative distributions {μs|s ∈ (0, 1)} on a free group F and study it as a whole. In this approach, the measure of a given set R ⊆ F is a function μ(R): s → μ s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

Original languageEnglish
Pages (from-to)705-731
Number of pages27
JournalInternational Journal of Algebra and Computation
Volume13
Issue number6
DOIs
StatePublished - Dec 2003

Keywords

  • Asymptotic density
  • Free groups
  • Generating functions
  • Growth functions
  • Measures
  • Negligible sets
  • Random walks
  • Recurrent and amenable groups
  • Regular sets
  • Sparse
  • Thick

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