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Fixed-point Proportion of Groups Acting on Rooted Trees

Sukkumnoed, Danthai LU (2026) In Bachelor's Theses in Mathematical Sciences MATK11 20261
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
We review groups acting on rooted trees and discuss their topological and measure-theoretic structure as compact topological groups using the theory of profinite groups and Haar measures.
We introduce new notions of fractal-like structures for groups acting on rooted trees: shifted fractality and shifted super strong fractality, as generalisations of fractality and super strong fractality.
These new notions generalise measure-theoretic results regarding the pushforward of measures and probabilistic independence.
Furthermore, we review the fixed-point process, fixed-point proportion, and application of martingale processes in this context.
We apply these theories to show that the fixed-point proportion of a level transitive shifted... (More)
We review groups acting on rooted trees and discuss their topological and measure-theoretic structure as compact topological groups using the theory of profinite groups and Haar measures.
We introduce new notions of fractal-like structures for groups acting on rooted trees: shifted fractality and shifted super strong fractality, as generalisations of fractality and super strong fractality.
These new notions generalise measure-theoretic results regarding the pushforward of measures and probabilistic independence.
Furthermore, we review the fixed-point process, fixed-point proportion, and application of martingale processes in this context.
We apply these theories to show that the fixed-point proportion of a level transitive shifted super-strongly fractal group acting on a bounded tree is zero; this gives a natural extension to the analogous result for super-strongly fractal groups. (Less)
Popular Abstract
Group theory is the study of symmetries of objects and their interactions.
The subject begins with the work of Galois in studying when a polynomial equation is solvable by radicals by exploiting the symmetries of solutions.
Later on, the development in group theory over the centuries brings immense applications in various fields, from the basis of cryptography to the spin structure of particles in quantum mechanics.

A group acting on a rooted tree is a group of symmetries of an infinite tree fixing a special vertex called the root.
This class of groups solves many longstanding open problems in group theory, such as Day's, Milnor's, and Burnside's problems.
In particular, we will study groups acting on rooted trees as topological... (More)
Group theory is the study of symmetries of objects and their interactions.
The subject begins with the work of Galois in studying when a polynomial equation is solvable by radicals by exploiting the symmetries of solutions.
Later on, the development in group theory over the centuries brings immense applications in various fields, from the basis of cryptography to the spin structure of particles in quantum mechanics.

A group acting on a rooted tree is a group of symmetries of an infinite tree fixing a special vertex called the root.
This class of groups solves many longstanding open problems in group theory, such as Day's, Milnor's, and Burnside's problems.
In particular, we will study groups acting on rooted trees as topological groups.
Roughly, we consider each symmetry on the infinite tree by stringing the same symmetry but only on the same tree with finitely many levels.
This gives a meaningful geometric and probabilistic structure to the group.

The main result of this thesis concerns the fixed-point proportions of groups acting on rooted trees.
This object captures the probability of finding a symmetry that fixes at least one infinite path on the whole tree.
We exploit the fractal-like structure of a certain class of groups, in which the symmetries in the group repeat on all the subtrees, to show that the fixed-point proportion is always zero when each vertex in the tree does not have too many descendants, and the group is shifted super-strongly fractal and level-transitive.
The fixed-point proportion is useful both within group theory and in complex and arithmetic dynamics, which have important applications in modelling dynamical systems that appear in ecology, economics, medicine, statistical mechanics, and so forth. (Less)
Please use this url to cite or link to this publication:
author
Sukkumnoed, Danthai LU
supervisor
organization
alternative title
Andelen fixpunkter av gruppverkningar på rotade träd
course
MATK11 20261
year
type
M2 - Bachelor Degree
subject
keywords
fixed-point proportion, groups acting on rooted trees, profinite groups, martingales
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4188-2026
ISSN
1654-6229
other publication id
2026:K6
language
English
id
9226080
date added to LUP
2026-05-18 14:04:30
date last changed
2026-05-18 14:04:30
@misc{9226080,
  abstract     = {{We review groups acting on rooted trees and discuss their topological and measure-theoretic structure as compact topological groups using the theory of profinite groups and Haar measures.
We introduce new notions of fractal-like structures for groups acting on rooted trees: shifted fractality and shifted super strong fractality, as generalisations of fractality and super strong fractality.
These new notions generalise measure-theoretic results regarding the pushforward of measures and probabilistic independence.
Furthermore, we review the fixed-point process, fixed-point proportion, and application of martingale processes in this context.
We apply these theories to show that the fixed-point proportion of a level transitive shifted super-strongly fractal group acting on a bounded tree is zero; this gives a natural extension to the analogous result for super-strongly fractal groups.}},
  author       = {{Sukkumnoed, Danthai}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Fixed-point Proportion of Groups Acting on Rooted Trees}},
  year         = {{2026}},
}