Advanced Seminar on Group Theory

Profinite Properties

Understanding infinite groups in general can be a very hard task. Looking only at finite groups should be a way easier task. Thus one might ask the question: What can one say about an infinite group by studing its finite images?

This can take multiple forms. For example the profinite rigidity of a group G ask, whether an arbitrary group H with the same finite quotiens as G is already isomorphic to G. Another question is that of profinite properties, i.e. whether a property of a group can be determined by the finite quotiens of a group: If G and H have the same finite quotients and G has some property (e.g. is abelian) does H also have to have the same property?

Both of these questions can be asked in an absolute sense, i.e. for general groups, or in a relative sense, i.e. for a certian class of groups, e.g. manifold groups, lattices in Lie groups, etc.

First we want to read the recent survey article Chasing finite shadows of inifinite groups through geometry by Bridson [1]. This will give us a some ideas of more geometric methods used to study groups by their finite quotients. Afterwards we will look at an explicit example: The paper Polycyclic groups, finite images, and elementary equivalence by Sabbagh and Wilson [2] where it is proven that being polycyclic is a profinite property.

While reading these two papers we will occasionally have mini-talks of 10–15 minutes shedding more light onto results/concepts to understand the main content. On demand, we can also offer seminar talks for students.

In the very end of the term this program might be rounded of by a research talk on more recent results in the field.

For further information please see the program.

References for the reading course:

1. M. Bridson, Chasing finite shadows of infinite groups through geometry, 2024. link
2. G. Sabbagh and J. Wilson, Polycyclic groups, finite images, and elementary equivalence, 1991. link