For a finitely generated group, the number of elements that can be spelled with words of length n, for any integer n>0, is called the growth function. This can be interpreted as a measure of the size of the group and is a powerful quasi-isometry invariant which has links to many areas of geometric group theory.
In the first lecture I will present the fundamental properties of the growth function and explore some key examples illustrating what kinds of functions can arise. I will also discuss Gromov’s important theorem on groups of polynomial growth.
In the second lecture I will discuss the formal power series associated to the growth function, which is known as the growth series. I will explain some ways in which the behaviour of the growth series can provide insight into the asymptotics, and demonstrate this with examples.
This video is part of the London Mathematical Society‘s Online Graduate Lecture Series. These are supported by the LMS, and organized by the North British Geometric Group Theory Seminar.
