Torsion-free Subgroups of GGS-Groups and Multi-EGS Groups

Abstract

In this thesis we use theory of branch groups and profinite groups to find infinite finitely generated residually finite torsion-free groups whose profinite completions are pro-p. We achieve this by defining a property analogous to the congruence subgroup property of integer matrix groups, which most GGS-groups satisfy, and use this to show that the commutator subgroups of such groups has pro-p profinite completion. Using elementary algebraic methods we find a simple criterion ensuring that these groups are also torsion-free which solves the problem. We also explore the application of this criterion to the more general class of multi-EGS groups where we find that a generalisation of the criterion can provide examples of multi-EGS groups with torsion-free commutator subgroup.

Popular Abstract

In mathematics we often want to find patterns and ways to generalise problems. At the heart of this lies abstract algebra, a field of mathematics focused on abstractly analysing all possible structures. In our work we focus on groups which are abstract objects that arise when you have some set of elements and some way of combining these elements. This could include simple structures such as that of addition or multiplication but also more complex things such as moves on a rubik’s cube (combined by doing one set of moves after another), matrix multiplication, or the set of symmetries of some geometrical object.

Groups have been studied extensively since the 19th and they come in many different flavours. Two main categories are finite groups, like the group of moves on a rubik's cube, and infinite groups such as the group of integers. A lot of things are known of finite groups but infinite groups have proven to be harder to analyse fully. As such we are often interested in infinite groups that are in some sense close to being finite but still manage to capture a lot of the infinite structure.

A particularly interesting collection of infinite groups that satisfy this are the so-called Grigorchuk-Gupta-Sidki groups, a family of groups that were found in the late 20th century and that consist of different ways of twisting a tree. These groups can be fully described in terms of just two elements but still manage to contain a lot of complex structure. Being easy to work with but still having rich structure has made GGS-groups a good place to look for groups with special properties and as such they have in recent years garnered much interest with applications found in areas ranging from cryptography and algebraic geometry to dynamics.

In this thesis we follow the work of Gustavo, Alejandra, and Jone who used GGS-groups to provide examples of groups satisfying a certain combinations of properties (specifically we want to find infinite, finitely generated, torsion-free, residually finite groups whose profinite completion is pro-p). We managed to use their methods to find additional examples of such groups and then extended some of the methods in the paper to a more general family of groups called multi-EGS groups where we found an analogous result.

• author: Engström, David
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Engineering)
• course: FMAM05 20241
• year: 2024
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: branch groups, GGS-groups, multi-EGS groups, profinite groups, pro-p groups, torsion-free groups, group theory
• publication/series: Master’s Theses in Mathematical Sciences
• report number: LUTFMA-3557-2024
• ISSN: 1404-6342
• other publication id: 2024:E66
• language: English
• id: 9173414
• date added to LUP: 2025-09-15 11:14:55
• date last changed: 2025-09-15 11:14:55