Automaton Groups And Self-Sımılar Groups


Thesis Type: Postgraduate

Institution Of The Thesis: Gazi Üniversitesi, Fen Bilimleri Enstitüsü, Turkey

Approval Date: 2019

Student: BEYZA NUR YAŞAR

Consultant: AYNUR ARIKAN

Abstract:

This thesis is about automata groups and self-similar groups. These groups are of interest because the class is the answer to some fundamental problems in group theory. The thesis consists of three main parts. In the first part, basic definition and information about rooted tree, tree automorphism, automorphism groups and automata are given. In addition, examples of the historical important automata groups are given in this part. In the second part deals with the algebraic properties of automata groups. In this part, the groups which are examples of the Burnside Problem related to periodic groups are examined. The main features of the Grigorchuk group and the Basilica group are discussed in detail. The last part is about the concept of growth in groups. The Milnor Problem related to the presence of intermediate growth groups is discussed. Intermediate growth groups that respond to this problem and whose first examples belong to the family of self-similar groups are discussed. Moreover, the relationship between amenable and self-similar groups is discussed and their role in answering the Day Problem is presented.