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Title: Branch groups and automata
Author: Wellen, George Arthur
ISNI:       0000 0004 2673 3309
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2008
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The focus of this thesis is finitely generated subgroups of the automorphism group of an infinite spherically homogeneous rooted tree (regular or irregular). The first chapter introduces the topic and outlines the main results. The second chapter provides definitions of the terminology used, and also some preliminary results. The third chapter introduces a group that appears to be a promising candidate for a finitely generated group of infinite upper rank with finite upper $p$-rank for all primes $p$. It goes on to demonstrate that in fact this group has infinite upper $p$-rank for all primes $p$. As a by-product of this construction, we obtain a finitely generated branch group with quotients that are virtually-(free abelian of rank $n$) for arbitrarily large $n$. The fourth chapter gives a complete classification of ternary automata with $C_2$-action at the root, and a partial classification of ternary automata with $C_3$-action at the root. The concept of a `windmill automaton' is introduced in this chapter, and a complete classification of binary windmill automata is given. The fifth chapter contains a detailed study of the non-abelian ternary automata with $C_3$-action at the root. It also contains some conjectures about possible isomorphisms between these groups.
Supervisor: Segal, Dan Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Group theory and generalizations (mathematics) ; Mathematics ; Automata ; Automaton