The Generation Problem in Thompson Group 𝐹

Abstract

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