Abstract
We show that the generation problem in Thompson’s 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
analogous 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 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 D and the construction of an
elementary amenable subgroup of F which is maximal in a normal subgroup of F.
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
analogous 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 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 D and the construction of an
elementary amenable subgroup of F which is maximal in a normal subgroup of F.
| Original language | English |
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| State | Published - 2016 |
Publication series
| Name | Arxiv preprint |
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Keywords
- math.GR