The intersection of subgroups in free groups and linear programming

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Abstract

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if H1 is a finitely generated subgroup of a free group F, then the WN-coefficient σ(H1) of H1 is rational and can be computed in deterministic exponential time in the size of H1. This coefficient σ(H1) is the minimal nonnegative real number such that, for every finitely generated subgroup H2 of F, it is true that r ¯ (H1, H2) ≤ σ(H1) r ¯ (H1) r ¯ (H2) , where r ¯ (H) : = max (r (H) - 1 , 0) is the reduced rank of H, r (H) is the rank of H, and r ¯ (H1, H2) is the reduced rank of the generalized intersection of H1 and H2. We also show the existence of a subgroup H2∗=H2∗(H1) of F such that r¯(H1,H2∗)=σ(H1)r¯(H1)r¯(H2∗), the Stallings graph Γ(H2∗) of H2∗ has at most doubly exponential size in the size of H1 and Γ(H2∗) can be constructed in exponential time in the size of H1.

Original languageEnglish (US)
Pages (from-to)1909-1940
Number of pages32
JournalMathematische Annalen
Volume370
Issue number3-4
DOIs
StatePublished - Apr 1 2018

Keywords

  • 20E07
  • 20F65
  • 90C90
  • Primary 20E05
  • Secondary 68Q25

ASJC Scopus subject areas

  • Mathematics(all)

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