The generation problem in Thompson group $F$

We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic fractions $\mathcal D$. Other applications of the study include the construction of new maximal subgroups of $F$ of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set $\mathcal D$ and the construction of an elementary amenable subgroup of $F$ which is maximal in a normal subgroup of $F$.