Engel Conditions in Branch Groups

By Thomas Smith //

This project sought to uncover the Engel conditions within the so-called ‘Twisted Twin of the Grigorchuk Group’. Mathematical groups can be understood as a set of ‘objects’ together with an operation that allows us to combine any two ‘objects’ within the group to obtain a third, along with numerous other criteria that distinguish it from a ‘set’. For example, the real number line with the operation of addition could be considered a group. One specific type of group is the family of branch groups. The first widely used example of a branch group was the Grigorchuk group, which can be thought of as a group that acts on the binary rooted tree – or as a group generated by symmetries of the tree in which each point has two branches incident from the point. We wanted to make a comparison between the Engel properties of the Grigorchuk group to determine whether it had any similarities with the Twisted Twin of the Grigorchuk group; a branch group discovered by Olivier Siegenthaler years later. In 2012, Laurent Bartholdi showed that the Grigorchuk group is not Engel, meaning that it does not satisfy the Engel property, i.e. it does not satisfy the Engel condition that the repeated commutator [[x,y],y],y]…,y] with n copies of y is trivial. We wanted to see how far this result could be generalised to the Twisted Twin.

In order to achieve this, I first had to understand the mathematical background surrounding groups of this nature. This was our first drawback, as it meant that we couldn’t initiate the research aspect of the project until I had a strong comprehension of the mathematical foundations that our results would be dependant upon. The two branch groups that were included in the report – the Grigorchuk group and its Twisted Twin – have a very different structure, so our main issues resonated with the computational factor of the report. These problems were overcome by finding ways to simplify the computations to make them easier to work with, such as taking the first level projections of words in the first level stabiliser of the Twisted Twin. This method allows us to shorten the words to make them easier to involve with the calculations. Another issue we found was working remotely from home. The content we were discussing during our meetings relied heavily on visual aspects, such as geometrical profiles and diagrams, and this heavily affected the speed at which I was learning the content needed to determine our results. The way in which we overcame this was by trialling different online meeting software to see which would be the best for sharing these diagrams.

The binary rooted tree

This experience has allowed me to develop my understanding of certain areas of mathematics I would have otherwise been denied if I had not been granted this incredible opportunity. I will use this to my advantage when I come to complete my PhD and hopefully secure a job in the research sector.

*To view Thomas’s research poster and presentation recording, please click on the thumbnails below: