Finite index subgroups of R. Thompsons group F

The authors classify the finite index subgroups of R. Thompson’s group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index subgroups of $F$ are isomorphic to $F$, and also separates the isomorphism classes of the finite index subgroups of $F$ which are not isomorphic to $F$ from each other; characterizing the structure of the extensions using properties of the structure of the finite index subgroups of $Z\times Z$.

💡 Research Summary

The paper delivers a complete classification of all finite‑index subgroups of R. Thompson’s group F. The authors begin by recalling that F is generated by two piecewise‑linear homeomorphisms of the unit interval and that its abelianisation is isomorphic to ℤ × ℤ. This observation underpins the whole analysis, because any finite‑index subgroup H ≤ F maps onto a finite cyclic quotient C via the natural projection onto the abelianisation, and the kernel of this projection is a finite‑index subgroup of the derived subgroup