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19.40 (2018)

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Does Thompson's group $F$ (see 12.20) have the Howson property, that is, is the intersection of any two finitely generated subgroups of $F$ finitely generated?

Contributor: I. Kapovich

Progress

No, it does not. Otherwise any subgroup of $F$ would have this property. But the wreath product $\mathbb{Z} \wr \mathbb{Z}$, which does not have the Howson property (A. S. Kirkinskiĭ, Algebra Logic, 20, no. 1 (1981), 24–36), can be embedded in $F$ (S. Cleary, Pacific J. Math., 228, no. 1 (2006), 53–61). (D. Robertson, Letter of 24 April 2018.)

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