Geometric Group Theory

Group theory was the first truly new concept at university. Mathematical analysis is off course also full of new ideas, but real numbers are informally known to everyone. Group theory is a suitable subject for demonstrating the power of abstraction. The following section is similar to the historical motivation of the subject. Let’s look at the addition of integers, multiplication of non-zero real numbers, shuffling cards and the symmetries of a square. Note that they have several properties in common. They are sets equipped with an operation (in case of the card the operation is “perform one shuffle, then the other”, similarly for a square). Moreover it has an identity element, which does nothing. For the examples above, the identity element is adding zero, multiplying by one, not moving the cards and not moving the square respectively. We can undo any change, after adding 5 add -5, after multiplying by 20 multiply by 1/20, sort cards after shuffle and return the square to its original position. We say that -5 is an additive inverse to 5.

We are now ready to make the second step towards abstraction. Let’s forget the motivating examples and keep just the structural properties. Let’s examine unspecified set which is equipped with an operation, identity element and inverses. Object with these properties is called a group. We lost a lot of information since we didn’t use all the properties of the motivating examples. For example unique prime factorisation of integers is not reflected in the structure of a group. However, advantages of this approach appear as soon as we try to prove something. It is easy with these new axioms to prove that the inverse to any element is unique and we don’t need to work with one group at a time.

The other component of the geometric group theory is topology, the science of shapes. An omnipresent question in topology is: “Can I deform this object to get the other one?” Topologist would know how to entangle Gordian knot. This subject is of interest to an algebraist as the group is naturally equipped with a topological structure. We can think of integers as a subset of real line, therefore identifying the elements of the group with point in a topological object, i.e. line. Given this set-up, we can further identify number 5 with a translation by 5 to the right. There is an analogous construction for a general group, where the elements of the group are identified with points and also with symmetries of this shape. In particular, such a shape exists for shuffling a deck of cards, multiplication of non-zero real numbers and the symmetries of a square.