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René Hartung

Publications and source records attributed to René Hartung.

8 recordsLinked to original sources
arXiv2012

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 finitely $L$-presented groups. Moreover, they allow us to prove that `being invariantly finitely $L$-presented' is an abstract property of a group which does not depend on the generating set. In the second part of this note, we consider finitely generated normal subgroups of finitely presented groups. Benli proved that a finitely generated normal subgroup of a finitely presented group is invariantly finitely $L$-presented whenever its quotient is infinite cyclic. We generalize this result to the case where the finitely presented group splits over its finitely generated subgroup and to the case where the quotient is abelian with torsion-free rank at most two.

math.GR↗
arXiv2012

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 groups can be described by a recursive (possibly infinite) presentation, a so-called finite $L$-presentation. Finite $L$-presentations allow numerous algorithms for finitely presented groups to be generalized to this special class of recursive presentations. We give an overview of the algorithms for finitely $L$-presented groups. As applications, we demonstrate how their implementation in a computer algebra system allows us to study explicit examples of self-similar groups including the Fabrykowski-Gupta groups. Our experiments yield detailed insight into the structure of these groups.

math.GR↗
arXiv2011

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 itself finitely $L$-presented. Our proof is constructive and it yields a finite $L$-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely $L$-presented group to be invariantly $L$-presented itself.

math.GR↗
arXiv2011

Coset enumeration for certain infinitely presented groups

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

math.GR↗
arXiv2011

Refined solvable presentations for polycyclic groups

We describe a new type of polycyclic presentations, that we will call refined solvable presentations, for polycyclic groups. These presentations are obtained by refining a series of normal subgroups with abelian sections. These presentations can be described effectively by presentation maps which yield the basis data structure to define a polycyclic group in computer-algebra-systems like {\scshape Gap} or {\scshape Magma}. We study refined solvable presentations and, in particular, we obtain consistency criteria for them. This consistency implementation demonstrates that it is often faster than the existing methods for polycyclic groups.

math.GR↗
arXiv2010

Solving linear equations over finitely generated abelian groups

We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $φ\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we discuss various algorithms for checking whether $b\in \imφ$ holds and if so, for computing a pre-image of $b$ in $G$ together with the kernel of $φ$.

math.GR↗
arXiv2007

A nilpotent quotient algorithm for L-presented groups

The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The appendix introduces two new infinite series of L-presented groups. Apart from being of interest in their own right, these new L-presented groups serve as examples for applications of the nilpotent quotient algorithm.

math.GR↗