Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients

@article{Hartung2010ApproximatingTS,
  title={Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients},
  author={Ren{\'e} Hartung},
  journal={Lms Journal of Computation and Mathematics},
  year={2010},
  volume={13},
  pages={260-271}
}
  • René Hartung
  • Published 1 December 2010
  • Mathematics
  • Lms Journal of Computation and Mathematics
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 multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski–Gupta groups, the Basilica group and the Brunner–Sidki–Vieira group. 

Tables from this paper

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

Refined solvable presentations for polycyclic groups

‎We describe a new type of presentation that‎, ‎when consistent‎, ‎describes a polycyclic group‎. ‎This presentation is obtained by‎ ‎refining a series of normal subgroups with‎ ‎abelian sections‎.

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

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

A Complete Bibliography of the LMS Journal of Computation and Mathematics

#P [20]. (2× 2 ·G): 2 [160]. 1 [174]. 12 [190]. 2 [154, 117, 172, 6, 51]. 3 [166]. 30 [11]. 4 [124]. 6 [142]. 6560 [54]. F4(q) [143, 161]. An [85]. B [73]. C3,4 [122]. d [198]. E6 [123]. E8 [123,

Coset Enumeration for certain Infinitely Presented Groups

TLDR
This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable.

References

SHOWING 1-10 OF 31 REFERENCES

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.

On residually torsion-free-nilpotent groups

Abstract Two examples of finitely presented soluble groups are constructed. The first is a metabelian residually torsion-free-nilpotent group G ≅ F/R having free central extension F/[F, R] which is

A just-nonsolvable torsion-free group defined on the binary tree

Abstract A two-generator torsion-free subgroup of the group of finite-state automorphisms of the binary tree is constructed having the properties of being just-nonsolvable and residually

The Twisted Twin of the Grigorchuk Group

TLDR
A twisted version of Grigorchuk's first group is studied, and it is shown that it admits a finite endomorphic presentation, has infinite-rank multiplier, and does not have the congruence property.

Endomorphic presentations of branch groups

On a Torsion-Free Weakly Branch Group Defined by a Three State Automaton

We study a torsion free weakly branch group G without free subgroups defined by a three state automaton which appears in different problems related to amenability, Galois groups and monodromy. Here...

Amenability via random walks

We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and ˙ Zuk. Our results separate the class of amenable groups from the closure of

On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are

Groups St Andrews 1997 in Bath

1. Radical rings and products of groups W. Amberg and Y. Sysak 2. Homogeneous integral table algebras of degrees two, three and four with a faithful element Z. Arad 3. Statistical methods in