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Nielsen equivalence classes of Grigorchuk-Gupta-Sidki groups

Vårdstedt Persson, Bror LU (2026) In Master's Theses in Mathematical Sciences MATM03 20261
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Mathematics (Faculty of Engineering)
Abstract
We consider the Nielsen equivalence classes and Andrews–Curtis
equivalence classes for finitely generated groups, specifically for the
Grigorchuk–Gupta–Sidki groups which act on rooted trees. We then
generalize a result of Myropolska to show that all torsion Grigorchuk–
Gupta–Sidki groups have infinitely many Nielsen equivalence classes
of generating pairs. Finally we deduce the number of Andrews–
Curtis classes of generating tuples of Grigorchuk–Gupta–Sidki groups
with non-constant defining vector.
Popular Abstract
Group theory has been formally studied since the nineteenth century and
is widely regarded as a large and fruitful subject of mathematics. Group
theory has applications in cryptography, physics, chemistry and various
other fields. Groups arise from studying the symmetries of symmetrical
objects, such as squares, polyhedra, or trees.
We consider a special family of groups called the Grigorchuk–Gupta–
Sidki groups. These describe the permutations of an infinitely branching
tree, and provide important examples of groups with exotic algebraic prop-
erties.
By using combinations of two specific permutations of the tree we can
generate the entire Grigorchuk–Gupta–Sidki group. We investigate how
many essentially different ways the... (More)
Group theory has been formally studied since the nineteenth century and
is widely regarded as a large and fruitful subject of mathematics. Group
theory has applications in cryptography, physics, chemistry and various
other fields. Groups arise from studying the symmetries of symmetrical
objects, such as squares, polyhedra, or trees.
We consider a special family of groups called the Grigorchuk–Gupta–
Sidki groups. These describe the permutations of an infinitely branching
tree, and provide important examples of groups with exotic algebraic prop-
erties.
By using combinations of two specific permutations of the tree we can
generate the entire Grigorchuk–Gupta–Sidki group. We investigate how
many essentially different ways the group can be generated by two generat-
ors. The concept of Nielsen equivalence classes is a formal way of partition-
ing up generating pairs that are essentially different. In this thesis we show
that a large subfamily of Grigorchuk–Gupta–Sidki groups have infinitely
many different such classes of generators. This is also the first extensive
family of groups with these properties. (Less)
Please use this url to cite or link to this publication:
author
Vårdstedt Persson, Bror LU
supervisor
organization
course
MATM03 20261
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Groups
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFMA-3167-2026
ISSN
1404-6342
other publication id
2026:E107
language
English
id
9250187
date added to LUP
2026-09-14 16:27:38
date last changed
2026-09-14 16:27:38
@misc{9250187,
  abstract     = {{We consider the Nielsen equivalence classes and Andrews–Curtis
equivalence classes for finitely generated groups, specifically for the
Grigorchuk–Gupta–Sidki groups which act on rooted trees. We then
generalize a result of Myropolska to show that all torsion Grigorchuk–
Gupta–Sidki groups have infinitely many Nielsen equivalence classes
of generating pairs. Finally we deduce the number of Andrews–
Curtis classes of generating tuples of Grigorchuk–Gupta–Sidki groups
with non-constant defining vector.}},
  author       = {{Vårdstedt Persson, Bror}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Nielsen equivalence classes of Grigorchuk-Gupta-Sidki groups}},
  year         = {{2026}},
}