Division algorithms in group rings of generalised hyperbolic groups

One of the most fundamental objects in mathematics is the integers, the set of whole numbers {...,-2, -1, 0, 1 ,2,...}, which are endowed naturally with the operations of addition and multiplication. Arguably the most powerful tool in the basic study of the integers is the Euclidean algorithm, which given two whole numbers n and m allows one to compute in an algorithmic way their greatest common divisor.

This algorithm implicitly underpins the majority of mathematical applications in modern technology, and is also a crucial part of the basic motivation for the field of algebraic number theory.

One of the key aims of this research project is to generalise this result to other rings seen commonly in topology and geometric group theory. These rings are called group rings, which are very useful for computing homology/cohomology of groups and topological spaces, as well as in representation theory. More recently in geometric group theory, group rings have allowed for major advances in the study of coherence, with a famous problem of Baumslag resolved by Jaikin-Zapirain-Linton in https://arxiv.org/abs/2303.05976, as well as being used by Kielak to give conditions for algebraic fibering in https://arxiv.org/abs/1809.09386 .

A recent paper of Avramidi-Delzant https://arxiv.org/abs/2309.16791 gave a powerful geometric construction of a division algorithm in group rings of groups acting in a certain way on hyperbolic spaces, including high genus surface groups. This algorithm allowed them to show that, for n depending on the action of the group, ideals of group rings generating by less than n elements are free, which has powerful topological and homological consequences.

In this project we hope to generalise this result in some sense to a larger class of groups acting in a less restrictive way on hyperbolic spaces. One interesting class of groups we hope to investigate are acyclindrically hyperbolic groups, which have a very general definition that includes mapping class groups and Out(F_n). We also hope to find more applications of these division algorithm results to problems of interest in the wider

field of geometric group theory, of which there is hope due to the recent success of the applications of group rings mentioned above.

This project falls within the EPSRC mathematical sciences research area, and fits into the wider ambitious work of the Oxford geometric group theory research team. Oxford university is a particularly strong location for this field of mathematics, and the wide variety of young up and coming researchers should provide ample opportunity for collaboration and discussion.