Chasing finite shadows of infinite groups through geometry

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 will sketch the rich history of this problem and describe how input from geometry and low-dimensional topology, mingling with algebra and arithmetic, have transformed the subject in recent years. I shall also describe some open problems.