On subgroups of R. Thompson's group $F$

Abstract

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 $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.Comment: 20 pages; v2: fixed some misprints, filled a gap in the proof of Theorem 4.1, added Remark 4.1 that Homeo^+(R) and many subgrioups of that group are quasi-residually finite; v3: Section 5 added, final version accepted to Transactions of the AM