New Implications of Arboreality and Hyperbolicity for Groups

A "group" is an mathematical concept that describes the notion of symmetry. For example, the collection of all ways of rotating a ball, fixing the centre, forms a group. It is possible to describe a group mathematically, without mentioning any particular object of which it is a collection of symmetries: it is a collection of things that you can combine to get something else in the collection, subject to a few rules. For example, the collection of all whole numbers, ..,-2,-1,0,1,2,..., is a group, because you can add two of them to get a new whole number.

Groups can have a large amount of mathematical structure, and there are many interesting questions about different aspects of groups, involving ideas from geometry, algebra, probability, and computing. For example, one can ask of a family of groups whether there is an algorithm --- an explicit procedure --- that decides whether two groups in the family are the same (this is harder than it sounds, because the same group can be described in many very different ways).

One particularly effective way to understand a group is to build some geometric object for which your group is a collection of symmetries. This makes the abstract group into something more concrete. Then, you study the object using geometry, and translate your conclusions into conclusions about the group. For example, if your group is the collection of whole numbers mentioned above, then a geometric object where the group is realised as a collection of symmetries could be a line: each whole number shifts you a specified distance along the line, in a way that is compatible with addition. For example, the number 3 corresponds to the symmetry that moves each point on the line 3 centimetres to the right, and -2 moves you 2 centimetres to the left. Their sum, 3-2=1, moves you 1 centimetre to the right, which is the same as first moving 3 to the right and 2 to the left. So, each member of the group is a symmetry of the line, and combining members of the group (in this case, adding numbers) corresponds to performing the symmetries in sequence. This is an example of what is called a "group action".

This project is in a part of mathematics called geometric group theory, where we study groups via their actions on geometric objects. The groups and geometries are generally more complicated than in the example, but the fundamental idea is the same.

The goal of the project is to understand two very large families of groups simultaneously. These families are defined by the property that the groups they contain are collections of symmetries of two types of spaces, each with a geometry that has many beautiful features. The project will investigate questions like: How are these two types of geometry related? If we make an algebraic change to the group, under what circumstances does the new group still exhibit the nice geometry? If there are many nice geometries one can associate to a fixed group, how do they all compare? Does the collection of all of these geometries itself exhibit interesting geometry? That last question seems abstract, but answering similar questions has proven very revealing in similar mathematical contexts in the past.

These questions come from pure mathematics, but the geometries involved are connected to other fields. For example, one of the types of space we will use is called a "median graph". Median graphs are deeply connected to certain problems in computer science involving breaking up tasks so that different computers can perform different pieces simultaneously, making a computation more efficient. Another type of geometry we will use can also be used in biology, to describe the possible ways that an organism could have evolved.

This project lies in a multi-disciplinary part of pure mathematics called "geometric group theory", and the aim is to study geometric structures with features reminiscent of negative curvature, and apply the conclusions to understand the structure of groups of symmetries of such spaces. The study of such "generalised negative curvature" groups is a very active part of geometric group theory at present, and the project will significantly build on theories that have already had major success.

The result will be a positive impact on other researchers in the field, who will be able to use the tools developed by the project in their own research. Since geometric group theory draws on many other parts of mathematics, it is possible that the proposed research will have technical consequences for other parts of mathematics, for example combinatorics. The proposed project will also benefit the UK mathematics research community through the training of a postdoctoral research assistant, who will develop their skills and knowledge by working on the project.

Mathematical history shows that ideas conceived for strictly mathematical reasons often have surprising scientific, societal, and economic consequences. While the specific questions the project will answer are in pure mathematics, and the immediate consequences of the answers will mainly be mathematical, it is possible that the tools developed could have consequences outside of pure mathematics. For example, the types of geometry that the project will study can also be used to model phenomena in other fields. Cubical geometry models the space of configurations of a robot, and CAT(0) geometry models the space of possible evolutionary ancestries of a biological organism. Finally, one of the most important geometries for this project is the geometry of hyperbolic spaces, which have various applications. For example, hyperbolic spaces can be used to model certain complex networks, including the Internet. Thus, it is possible that a better understanding of these abstractly-defined geometries could have concrete effects in other scientific areas.