# Symmetry in Newcastle

## Monday, 25th Jan 2021

Schedule (Zoom):

18.30-19.30: François Le Maître
19.30-20.00: Break
20.00-21.00: Charles Cox

# Dr François Le Maître

(Institut de Mathématiques de Jussieu–PRG, Université de Paris)

# Dense totipotent free subgroups of full groups

In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on $n>1$ generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means "as rich as possible", and furthermore that the free group can be made dense in the ambient full group of the equivalence relation. This is joint work with Alessandro Carderi and Damien Gaboriau.

# Dr Charles Cox

(School of Mathematics, University of Bristol)

My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups (in https://arxiv.org/abs/1907.05498). A group $G$ has spread $k$ if for every $g_1,\ldots,g_k$ we can find an $h$ in $G$ such that $\langle g_i, h\rangle=G$. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread $0$. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven’s first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we’ll see the answer to this question. The arguments will be concrete (*) and accessible to a general audience.