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 mid20th 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 prop 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 socalled finite pgroups. The main aspect that I will work on is the Hausdorff dimension of prop 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 noninteger 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 prop groups. Just as the Hausdorff dimension of a fractal gives an indication as to how much space the fractal takes up in the 2dimensional plane, for a given prop 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 longstanding open question: Let G be a finitely generated prop group with finite Hausdorff spectrum. Does this imply that G is padic analytic? Here a padic analytic prop group is a prop group with rich geometric structure.
Barnea and Shalev proved in 1997 that if G is a padic analytic prop group, then the Hausdorff spectrum is finite. Hence if we establish our goal, this would give a much sought after invariant for prop groups that are not padic analytic. As an application, our result would then deliver a concrete method of identifying when a prop group is padic analytic or not.
Regarding our goal, the initial approach is to look at prop groups that are close to being padic analytic, such as the group constructed by JaikinZapirain and Klopsch in 2007. This group constructed by JaikinZapirain and Klopsch is not padic analytic, but has properties that padic 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.
