 SYMMETRY IN NEWCASTLE
 Location: (Online Campus)
 Dates: Fri, 7^{th} Aug 2020  Fri, 7^{th} Aug 2020
 Speaker: Prof Anthony Guttmann, Department of Mathematics and Statistics, The University of Melbourne
 Title: On the amenability of Thompson's Group $F$
 Abstract for On the amenability of Thompson's Group $F$:
In 1967 Richard Thompson introduced the group $F$, hoping that it was nonamenable, since then it would disprove the von Neumann conjecture.
Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I
will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including
mention of the many incorrect "proofs" of amenability or nonamenability. Then I will give details of a new, efficient algorithm for obtaining terms of the
cogrowth sequence. Finally I will describe a number of numerical methods to analyse the cogrowth sequences of a number of infinite, finitelygenerated groups,
and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an
alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price).
 Speaker: Dr Collin Bleak, University of St Andrews
 Title: On the complexity of elementary amenable subgroups of R. Thompson's group $F$
 Abstract for On the complexity of elementary amenable subgroups of R. Thompson's group $F$:
The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly wellordered by the embeddability relation in type $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a ﬁnitely generated elementary amenable subgroup of F whose EAclass is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed.
Joint work with Matthew Brin and Justin Moore.
 Abstract for On the amenability of Thompson's Group $F$:
In 1967 Richard Thompson introduced the group $F$, hoping that it was nonamenable, since then it would disprove the von Neumann conjecture.
Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I
will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including
mention of the many incorrect "proofs" of amenability or nonamenability. Then I will give details of a new, efficient algorithm for obtaining terms of the
cogrowth sequence. Finally I will describe a number of numerical methods to analyse the cogrowth sequences of a number of infinite, finitelygenerated groups,
and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an
alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price).
 Abstract for On the complexity of elementary amenable subgroups of R. Thompson's group $F$:
The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly wellordered by the embeddability relation in type $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a ﬁnitely generated elementary amenable subgroup of F whose EAclass is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed.
Joint work with Matthew Brin and Justin Moore.
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