Coarse Geometry of Groups and Spaces

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A "group" is a family of symmetries of a geometric object. The subject of this research - geometric group theory - seeks to reveal the connections between the geometric properties of the object and the algebraic properties of its group of symmetries.From the very beginning, group theory has been driven by its applications in mathematics and beyond, including: Galois' original theory on the roots of polynomials, Noether's theorem relating symmetries of a physical system to its' conservation laws, Lie and Kac-Moody groups in physics, crystallography in chemistry and material science, and public-key cryptography. Group theory enjoys dynamic interactions with computer science, particularly in the advancing fields of AI and machine learning; and shapes our understanding of topological spaces, geometry and number theory. Many natural geometric properties (for example growth, dimension and curvature) have been intensively studied for their algebraic consequences. To give two examples, groups with polynomial growth are completely algebraically explained by Gromov's remarkable Polynomial Growth Theorem, there are many families of groups with exponential growth. Much more recently, examples of groups with "intermediate growth" were discovered, these groups are deeply mysterious. Another of Gromov's remarkable contributions is to show that a randomly chosen group is almost surely hyperbolic: that is, it is the group of symmetries of a geometric space with negative curvature.A weakness of this approach is that many of these geometric properties only pass from highly symmetric geometric objects to nearly complete families of their symmetries - formally the group and the geometry are quasi-isometric, meaning they appear "the same" at sufficiently large scales. To study the more general relationship we need to allow the possibility that the group sits inside the geometric object in a highly-distorted way. Of the multitude of invariants known to geometric group theory, very few behave sufficiently well in this more general setting to give productive results.My proposal concerns a new family of distortion-proof (coarse) invariants called Poincaré profiles, which I recently introduced. Poincaré profiles essentially measure how robustly connected parts of a geometric space can be on a variety of different scales. I have established that there is a connection between the Poincaré profiles of a hyperbolic group and the (conformal) dimension of a fractal associated to the group. Fractals are intricate shapes which exhibit self-similarity at increasingly small scales and there can be many different yet sensible ways to measure their dimension - none of which is necessarily an integer. One source of fractals is from boundaries of hyperbolic groups: visualisations of that group as seen "from infinity". A key goal of my proposal is to exactly reveal this relationship to improve our understanding of both hyperbolic groups and fractals.More generally, there is a great need for further coarse invariants. Many structural results in geometric group theory are likely to have natural coarse analogues if one can find the right invariants. I have many ideas of problems which can be dealt with using new invariants I will define inspired by tools from analysis, algebraic topology, combinatorics, computer science and theoretical physics. There are also many natural applications of this work, since finding and quantifying well-connected parts of a network is a common goal in advertising algorithms, geometric deep learning, protein interaction modelling and graph neural networks. The continued development and improvement of these techniques has industrial and societal benefits ranging from improved financial forecasting and better 3D facial and speech recognition, to more accurate and efficient drug design and composite material design and testing.