Talks

Talks and Abstracts

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Plenary Speakers

We are honoured to have as our invited speakers this year:

Growth of conjugacy classes in finite groups

Let $K$ be a conjugacy class in a finite group and $K^n = \{ x_1 \ldots x_n: x_i \in K \}$. We consider the sequence $\{ |K^n| \}$, and the implications of this seqeunce on the algebraic structure of $G$.

An homage to homogeneity

An object X in a category C is called HOMOGENEOUS if the existence of a local symmetry inside X implies the existence of a global symmetry on X. If G is the automorphism group of such an object, then it necessarily satisfies strong transitivity properties.

We will briefly explore some well-known categories where homogeneous objects are two a penny... and where the associated automorphism groups are familiar and well-understood.

We will then turn our attention to the problem of finding homogeneous objects for a given automorphism group. To do this we will need to work in the category of RELATIONAL STRUCTURES. We will discuss some lovely results from model theory that apply to this problem and which give us a framework through which to view the universe of finite permutation groups.

Profinite rigidity of lattices in higher rank Lie groups

A typical first approach to an infinite group is to study its finite quotients. To understand the limits of this approach it is fruitful to find examples of groups which are really different (say non-commensurable) but have the same finite quotients. In this spirit we discuss which simple Lie groups admit uniform lattices (i.e., discrete cocompact subgroups) which are not commensurable but whose profinite completitions are commensurable. We explain why such examples can be found in most simple Lie groups of higher rank. However, more surprisingly, we indicate why in some exceptional Lie groups such lattices cannot exists. (Based on joint work with H. Kammeyer).

An irrational slope Thompson's group

In this talk I will discuss a relative to Thompson's group $F$, the group $F_\tau,$ which is the group of piecewise linear homeomorphisms of $[0,1]$ with breakpoints in $\mathbb{Z}[\tau]$ and slopes powers of $\tau,$ where $\tau = \frac{\sqrt5 -1}{2}$ is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type $\operatorname{F}_\infty.$ Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to find an explicit presentation and the abelianization of the group. The surprising feature is that the $T$- and $V$-versions of these groups are not simple, however. This is joint work with J. Burillo and L. Reeves. If time permits I will say a few words about the generalisations studied by N. Winstone.

Contributed Talks

These are 20-25 minute talks, with an additional 5 minutes for questions. A full list of talks can be found in our booklet.