Coset Enumeration for certain Infinitely Presented Groups

@article{Hartung2011CosetEF,
  title={Coset Enumeration for certain Infinitely Presented Groups},
  author={Ren{\'e} Hartung},
  journal={Int. J. Algebra Comput.},
  year={2011},
  volume={21},
  pages={1369-1380}
}
  • René Hartung
  • Published 1 June 2011
  • Mathematics
  • Int. J. Algebra Comput.
We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group and the Hanoi 3-group 
A Reidemeister-Schreier theorem for finitely $L$-presented groups
We prove a variant of the well-known Reidemeister-Schreier theorem for finitely $L$-presented groups. More precisely, we prove that each finite index subgroup of a finitely $L$-presented group is
Computability of finite quotients of finitely generated groups
Abstract We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular
Subgroup induction property for branch groups.
Recently, the so-called subgroup induction property attracted the attention of mathematicians working with branch groups. It was for example used to prove that groups with this property are subgroup
A Note on Invariantly Finitely $L$-Presented Groups
In the first part of this note, we introduce Tietze transformations for $L$-presentations. These transformations enable us to generalize Tietze's theorem for finitely presented groups to invariantly
Investigating self-similar groups using their finite $L$-presentation
Self-similar groups provide a rich source of groups with interesting properties; e.g., infinite torsion groups (Burnside groups) and groups with an intermediate word growth. Various self-similar
Presentations and Structural Properties of Self-similar Groups and Groups without Free Sub-semigroups
This dissertation is devoted to the study of self-similar groups and related topics. It consists of three parts. The first part is devoted to the study of examples of finitely generated amenable
On n-Engel pair satisfying certain conditions
Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,n g], g = [g,n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh-2gh = ghg

References

SHOWING 1-10 OF 27 REFERENCES
A finitely generated, infinitely related group with trivial multiplicator
  • G. Baumslag
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1971
We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator. 1. The purpose of this note is to establish the exitense of a finitely generated group which
Determining Subgroups of a Given Finite Index in a Finitely Presented Group
The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally
A Nilpotent Quotient Algorithm for Certain Infinitely Presented Groups and its Applications
TLDR
A nilpotent quotient algorithm is described for a certain class of infinite presentations: the so-called finite L-presentations and conjectural descriptions of the lower central series structure of various interesting groups including the Grigorchuk supergroup, the Brunner–Sidki–Vieira group, the Basilica group, and certain generalizations of the Fabrykowski–Gupta group are obtained.
Non-amenable finitely presented torsion-by-cyclic groups
Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an
Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients
We describe an algorithm for computing successive quotients of the Schur multiplier M ( G ) of a group G given by an invariant finite L -presentation. As applications, we investigate the Schur
A practical method for enumerating cosets of a finite abstract group
An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relations between k generating operations S 1 , …., S k of , such that
ON THE FAILURE OF THE CO-HOPF PROPERTY FOR SUBGROUPS OF WORD-HYPERBOLIC GROUPS
We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic group G such that H is one-ended, and H does not split over a cyclic group, and H is isomorphic to one of its
Computation with finitely presented groups
  • C. Sims
  • Mathematics
    Encyclopedia of mathematics and its applications
  • 1994
1. Basic concepts 2. Rewriting systems 3. Automata and rational languages 4. Subgroups of free products of cyclic groups 5. Coset enumeration 6. The Reidemeister-Schreier procedure 7. Generalized
Proving a group trivial made easy: A case study in coset enumeration
Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups. The process is not well understood, and various
The congruence subgroup problem for branch groups
We state and study the congruence subgroup problem for groups acting on rooted trees, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we
...
...