Non-Elementary Amenable Groups. Kaplansky's Conjecture

Chornyi, Maksym

doi:https://doi.org/10.21985/n2-80yy-c992

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Non-Elementary Amenable Groups. Kaplansky's Conjecture

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This dissertation addresses the property of amenability of discrete groups and their actions. In Chapter 2, following the introduction, all necessary definitions are given to introduce amenable groups, elementary amenable groups, random walks, topological full groups, Thompson's group $F$ and to show connections between them. The chapter also briefly covers the history of the topic including several open questions. In Chapter 3, new examples of simple finitely generated groups arising from actions of free abelian groups on the Cantor sets are constructed. As particular examples, interval exchange transformation groups and a group naturally associated with the Penrose tilings are discussed. Many of these groups are non-elementary amenable. Chapter 4 discusses Thompson's group $F$. The primary goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group $F$ using certain properties of extensive amenability. While no conclusive answer regarding the amenability of $F$ is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions. Chapter 5 provides additional insight into the topic of Chapter 4 and contains some observations related to random walks which do not fit directly into the context of the chapter. In particular, section 5.1 gives an example of a non-extensively amenable group action whose amenability properties can be conveniently checked using the technique developed in Chapter 4. Chapter 6 discusses Kaplansky's zero-divisor conjecture, another open question in group theory not directly related to the previous chapters. We describe a computer algorithm, based on a previously published paper, which makes it possible to efficiently prove the conjecture for some partial cases.

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