Aspects of the topological dynamics of sparse graph automorphism groups
File(s)
Author(s)
Sullivan, Robert
Type
Thesis
Abstract
We examine sparse graph automorphism groups from the perspective
of the Kechris-Pestov-Todorčević (KPT) correspondence. The sparse
graphs that we discuss are Hrushovski constructions: we consider the
'ab initio’ Hrushovski construction M_0, the Fraïssé limit of the class of 2-sparse graphs with self-sufficient closure; M_1, a simplified version
of M_0; and the ω-categorical Hrushovski construction M_F. We prove
a series of results that show that the automorphism groups of these
Hrushovski constructions demonstrate very different behaviour to previous classes studied in the KPT context. Extending results of Evans,
Hubička and Nešetřil, we show that Aut(M_0) has no coprecompact
amenable subgroup. We investigate the fixed points on type spaces
property, a weakening of extreme amenability, and show that for a
particular choice of control function F, Aut(M_F) does not have any
closed oligomorphic subgroup with this property. Next we consider the
Aut(M_1)-flow of linear orders on M_1, and show that minimal subflows
of this have all Aut(M_1)-orbits meagre. We give partial analogous results for the Aut(M_0)-flow of linear orders on M_0, and find the universal
minimal flow of the automorphism group of the “dimension 0” part of
M_0.
of the Kechris-Pestov-Todorčević (KPT) correspondence. The sparse
graphs that we discuss are Hrushovski constructions: we consider the
'ab initio’ Hrushovski construction M_0, the Fraïssé limit of the class of 2-sparse graphs with self-sufficient closure; M_1, a simplified version
of M_0; and the ω-categorical Hrushovski construction M_F. We prove
a series of results that show that the automorphism groups of these
Hrushovski constructions demonstrate very different behaviour to previous classes studied in the KPT context. Extending results of Evans,
Hubička and Nešetřil, we show that Aut(M_0) has no coprecompact
amenable subgroup. We investigate the fixed points on type spaces
property, a weakening of extreme amenability, and show that for a
particular choice of control function F, Aut(M_F) does not have any
closed oligomorphic subgroup with this property. Next we consider the
Aut(M_1)-flow of linear orders on M_1, and show that minimal subflows
of this have all Aut(M_1)-orbits meagre. We give partial analogous results for the Aut(M_0)-flow of linear orders on M_0, and find the universal
minimal flow of the automorphism group of the “dimension 0” part of
M_0.
Version
Open Access
Date Issued
2021-09
Date Awarded
2022-01
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Evans, David
Sponsor
Imperial College London
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)