Normal subgroups of the automorphism groups of some homogeneous structures
File(s)
Author(s)
Li, Yibei
Type
Thesis
Abstract
We will study the normal subgroup structures of the automorphism groups of some homogeneous structures in this thesis. In particular, we prove the simplicity of the automorphism groups of the following homogeneous structures:
(i) all of the symmetric ‘semi-free’ homogeneous structures described by
Cherlin in the appendix of [Che98],
(ii) some of the asymmetric ‘semi-free’ homogeneous structures in the appendix of [Che98],
(iii) the linear order expansions of non-trivial free homogeneous structures,
and
(iv) the universal n-linear order for n ≥ 2.
The methods used generalise techniques developed in a sequence of papers by Macpherson and Tent [MT11] and Tent and Ziegler [TZ13a], [TZ13b]. A key notion in these methods is that of a stationary independence relation on finite substructures of a homogeneous structure M defined in [TZ13a]. This generalised the idea of substructures of M being freely amalgamated over their intersection if M is free, used in [MT11].
A stationary independence relation is symmetric. We can apply Tent and Ziegler’s method in [TZ13a] directly to the symmetric semi-free homogeneous structures in (i). But we cannot do so to the structures in (ii)-(iv) as they are asymmetric. One of the main contributions of our work is to drop the
symmetry requirement, giving us the notion of a stationary weak independence relation. We then show how this can be used to prove the simplicity of the automorphism groups of the structures in (ii)-(iv) following a similar method of [MT11] and [TZ13a].
(i) all of the symmetric ‘semi-free’ homogeneous structures described by
Cherlin in the appendix of [Che98],
(ii) some of the asymmetric ‘semi-free’ homogeneous structures in the appendix of [Che98],
(iii) the linear order expansions of non-trivial free homogeneous structures,
and
(iv) the universal n-linear order for n ≥ 2.
The methods used generalise techniques developed in a sequence of papers by Macpherson and Tent [MT11] and Tent and Ziegler [TZ13a], [TZ13b]. A key notion in these methods is that of a stationary independence relation on finite substructures of a homogeneous structure M defined in [TZ13a]. This generalised the idea of substructures of M being freely amalgamated over their intersection if M is free, used in [MT11].
A stationary independence relation is symmetric. We can apply Tent and Ziegler’s method in [TZ13a] directly to the symmetric semi-free homogeneous structures in (i). But we cannot do so to the structures in (ii)-(iv) as they are asymmetric. One of the main contributions of our work is to drop the
symmetry requirement, giving us the notion of a stationary weak independence relation. We then show how this can be used to prove the simplicity of the automorphism groups of the structures in (ii)-(iv) following a similar method of [MT11] and [TZ13a].
Version
Open Access
Date Issued
2020-09
Date Awarded
2021-04
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Evans, David
Sponsor
Imperial College London
Engineering and Physical Sciences Research Council
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)