Chasing finite shadows of infinite groups through geometry

This is the video of the MATH+ Friday lecture by Martin R. Bridson (U Oxford) entitled "Chasing finite shadows of infinite groups through geometry". The talk was given online on May 21, 2021.

ABSTRACT:

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite Asynch Comb 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, Bridson will sketch some of the rich history of this problem. He will discuss how hard it is to decide which groups have finite images, connecting this problem to the structure of finite simple groups, and describe how in- put from geometry and low-dimensional topology, mingling with algebra and arithmetic, have transformed the subject in recent years. He will then explain why certain groups arising in 2-D and 3-D hyperbolic geometry can be distinguished from all other groups using only their finite images and eventually highlight some open problems.