It is a natural question to ask how much algebraic information is encoded in the set of finite quotient of a given group. More precisely, one tries to establish which properties of infinite, discrete, residually finite groups are preserved under isomorphisms of their profinite completions. A group is said to be (absolutely) profinitely rigid if its isomorphism type is completely determined by its profinite completion. The first talk will focus on the history of this problem, covering some classical results as well as more recent work and open problems in the area. We will introduce all the necessary background, so no prior knowledge of the topic will be assumed.

A variation of this problem involves restricting to a certain family of groups and trying to decide whether a group is profinitely rigid relative to this family. Much work has been done towards solving this problem for fundamental groups of 3-manifolds. In the second talk, we will focus our attention on a related family of groups known as free-by-cyclic groups, which have natural connections with 3-manifolds. We will see that many properties of free-by-cyclic groups are invariants of their profinite completion. As a consequence, we obtain various profinite rigidity results, including the almost profinite rigidity of generic free-by-cyclic groups amongst the class of all free-by-cyclic groups. 

This is joint work with Sam Hughes. 

Join Zoom meeting ID 613 8691 2732

Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)