Profinite rigidity: Chasing finite shadows of infinite groups through geometry

There are situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects – i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images (equivalently, its profinite completion)? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of a given infinite group are? I will sketch some of the rich history of these problems and present results that illustrate how the field surrounding these questions has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.

You can learn more about Professor Martin Bridson here.