The generation problem in Thompson group $F$

Abstract

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 $F$.

85 pages, final version, to appear in Memoirs of the AMS

References in corpus (1)

• Finite index subgroups of R. Thompson's group F

Cited by in corpus (7)

• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• On Jones Subgroup of R. Thompson's Group $T$
• An introduction to Thompson knot theory and to Jones subgroups
• Divergence of Thompson groups
• Invariable generation of Thompson groups
• On closed subgroups of the R. Thompson group $F$
• Random Generation of Thompson's group $F$