Cohomology and Actions of Groups

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A very classical thing to study, going back to the Ancient Greeks, is the symmetries of an object. In the abstract these are known as groups; they can be very complicated and difficult to understand. There are many ways of investigating these abstract groups of symmetries and trying to impose some order on them; one of these is to associate to the group a new algebraic invariant, of which cohomology is a prime example. However, these new invariants can be very difficult to understand, or even to calculate, partly because they are infinite. By using a quantity called the regularity it is possible to show that these infinite objects are controlled by a specific finite part of themselves and they then become much easier to deal with. This project aims to make these techniques available for much larger classes of groups than before and, in so doing, obtain much more detailed information about the structure of these groups and their cohomology.