Aspects of Growth in Infinite Groups
Lead Research Organisation:
Heriot-Watt University
Department Name: S of Mathematical and Computer Sciences
Abstract
We will study various kinds of growth in infinite groups, with particular focus on the growth of conjugacy classes. Benson's 1983 proof that the standard growth series of a virtually abelian group is rational with respect to any generating set introduced the idea of partitioning the language of words over the generators into 'patterned sets', which allows a linear algebraic approach to growth. The same paper also provides a criterion for growth to be rational. We hope to make use of these ideas to investigate the growth of conjugacy classes in virtually abelian groups. Other areas of investigation include growth of cosets, and relative subgroup growth.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509474/1 | 01/10/2016 | 30/09/2021 | |||
| 1820554 | Studentship | EP/N509474/1 | 01/10/2016 | 20/04/2020 | Alexander Evetts |
| Description | I have succeeded in proving that finitely generated virtually abelian groups have rational conjugacy growth series, rational coset growth series (with respect to any subgroup), and that arbitrary subgroups of virtually abelian groups have rational relative growth series. The notion of growth of a finitely generated group (which is a certain kind of algebraic structure) is one the classical strands of the modern field of Geometric Group Theory. Growth in this context is a measure of how quickly the group gets large. This can be studied via geometry, combinatorics, algebra, formal language theory, and other approaches. In this research, I have built on and adapted work of M Benson in 1983 to prove certain 'regularity' results about the growth of virtually abelian groups. In order to do this I generalised Benson's notion of 'patterns' to a notion of 'd-fold patterns', which facilitate counting the growth of finite tuples of objects, rather than single objects. I have submitted a paper to the Illinois Journal of Mathematics and I am awaiting a reply. In additional to the URL provided below, I wrote a 'Microthesis' for the newsletter of the London Mathematical Society: https://www.lms.ac.uk/sites/lms.ac.uk/files/files/NLMS_479_for%20web.pdf. Update 10/03/2020: The paper mentioned above on virtually abelian groups has been accepted and published in the Illinois Journal of Mathematics: Illinois J. Math. 63 (2019), no. 4, 513-549. Together with my supervisor (Ciobanu) and another collaborator (Ho) we have shown that the conjugacy growth series of the soluble Baumslag-Solitar groups is transcendental (for the standard generating sets). This result used a combination of new and existing combinatorics, and formal language theory. We have written a paper that has been accepted for publication in the New York Journal of Mathematics. I have been increasing my knowledge of the existing body of work on conjugacy growth in nilpotent groups, and I have made some incremental contributions. These will form a chapter of my thesis. |
| Exploitation Route | One aspect of these findings (namely that the conjugacy growth of virtually abelian groups is rational) contributes part of the answer to the following conjecture: a group has rational conjugacy growth if and only if it is finite, or virtually abelian. It is also hoped that the techniques of patterns and polyhedral sets may prove useful when considering virtually nilpotent groups, which can be thought of as a generalisation of virtually abelian groups. |
| Sectors | Other |
| URL | https://www.macs.hw.ac.uk/~ace2/ |