Group theory, the study of symmetries, is a branch of pure mathematics that began in the early 19th century, but it was really only from the mid-20th century when the importance of groups went beyond pure mathematics to other sciences, and indeed the real world. Without group theory, there would be no computers, and matters involving cryptography and security issues would be severely stunted. Time and time again, the influence of groups to applied mathematics and physics has been confirmed. Just as mathematics is the queen of all sciences, advancements in pure mathematics are vital to the advancement in applied mathematics.
Now a group is just a collection of symmetries, for instance, the rotations of a cube about one of its central axes. In group theory, these symmetries are often studied abstractly and independent of the object on which they operate. This allows us to generalise and strengthen results.
My proposed research concerns a class of infinite groups, called pro-p groups. These are infinite groups made up of finite pieces that are strung together in a coherent way. These finite pieces are finite groups of prime power size, the so-called finite p-groups. The main aspect that I will work on is the Hausdorff dimension of pro-p groups.
Hausdorff dimension was first applied in the 1930s to fractals and shapes in nature, as a generalisation of our usual concept of integer dimension (for instance, a line has dimension 1, a disc has dimension 2 and a solid cube has dimension 3) to non-integer dimension. This generalised dimension played a key role in explaining the coastline paradox, which is the phenomenon that the more accurate the measuring instrument, the longer the coastline becomes. Mandelbrot showed in 1967 that the Hausdorff dimension of a coastline is strictly between 1 and 2. In other words, a coastline takes up more space than a line, but has no area.
Hausdorff dimension was initially constrained to fractals and sets in our standard space. But in the 1990s, Abercrombie, Barnea and Shalev applied Hausdorff dimension to certain kinds of infinite groups, namely pro-p groups. Just as the Hausdorff dimension of a fractal gives an indication as to how much space the fractal takes up in the 2-dimensional plane, for a given pro-p group G, the Hausdorff dimension of a subgroup H tells us how dense the subgroup H is in G. It is of interest to study the Hausdorff dimensions of all subgroups in a given group; in other words, the Hausdorff spectrum. This Hausdorff spectrum, which is a subset of the unit interval [0,1], encodes information on the density spread of subgroups within a given group.
My main goal is to answer the following long-standing open question: Let G be a finitely generated pro-p group with finite Hausdorff spectrum. Does this imply that G is p-adic analytic? Here a p-adic analytic pro-p group is a pro-p group with rich geometric structure.
Barnea and Shalev proved in 1997 that if G is a p-adic analytic pro-p group, then the Hausdorff spectrum is finite. Hence if we establish our goal, this would give a much sought after invariant for pro-p groups that are not p-adic analytic. As an application, our result would then deliver a concrete method of identifying when a pro-p group is p-adic analytic or not.
Regarding our goal, the initial approach is to look at pro-p groups that are close to being p-adic analytic, such as the group constructed by Jaikin-Zapirain and Klopsch in 2007. This group constructed by Jaikin-Zapirain and Klopsch is not p-adic analytic, but has properties that p-adic analytic groups possess. The first step would be to compute the Hausdorff dimensions of all closed subgroups of this group G. If this Hausdorff spectrum is finite, then the problem is solved in the negative. If the finitely generated Hausdorff spectrum of this group is in infinite, then this provides evidence that the answer to the problem should be yes.
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