GAG 2019

"Chasing finite shadows of infinite groups"

There are many situations in geometry and group theory where it is natural, convenient or necessary
to explore infinite groups via their actions on finite objects. But how much understanding can one really gain about an infinite group by examining its finite images? Sometimes little, sometimes a lot.

In this colloquium talk, I shall sketch the rich history of this problem and describe how input from geometry and low-dimensional topology have transformed the subject in recent years. In particular I shall explain how McReynolds, Reid, Spitler and I settled one of the outstanding problems in the field with a mixture of algebra, arithmetic and 3-dimensional hyperbolic geometry.