On subgroups of R. Thompson’s group F

We provide two ways to show that the R. Thompson group F has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of F on (0, 1), thus solving a problem by D. Savchuk. The first way employs Jones’ subgroup of the R. Thompson group F and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings’ core graphs and gives many implicit examples. We also show that F has a decreasing sequence of finitely generated subgroups F >H₁ > H₂ > ··· such that ∩H_i = {1} and for every i there exist only finitely many subgroups of F containing H_i.