# Alan Reid - Virginia Mathematics Lectures - October 28-30, 2024

#### Event start date: Monday, October 28, 2024

Virginia Mathematics Lectures »
### Finite shadows of infinite groups: What do they hide?

- Lecture 1: Monday, October 28, 2024,
*5-6PM, Wilson 402* - **Exploring infinite groups by finite quotients**
- Lecture 2: Tuesday, October 29, 2024,
*5-6PM, Wilson 402* - **What is profinite rigidity?**
- Lecture 3: Wednesday, October 30, 2024,
*5-6PM, Wilson 402* - **Grothendieck’s Problem and detecting finitely generated versus finitely presented**

## Lecture 1: Exploring infinite groups by finite quotients

**Abstract:** 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 — ie via their finite quotients. A natural question is therefore where do we find finite quotients, and how can we use them to distinguish infinite groups. Lecture 1 will provide an elementary discussion of these topics.

## Lecture 2: What is profinite rigidity?

**Abstract:** The set of finite quotients of a finitely generated group is neatly captured by its profinite completion. A finitely generated (residually finite) group G is called profinitely rigid if whenever another finitely generated (residually finite) group H has isomorphic profinite completion, then H is isomorphic to G. This talk will discuss some history and some recent progress on groups that are (are not) profinitely rigid. In particular I will discuss results connected to the study of 3-manifold groups.

## Lecture 3: Grothendieck's Problem and detecting finitely generated versus finitely presented

**Abstract:** A particular construction of groups that fail to be profinitely rigid, arise from Grothendieck Pairs (G,H) where H is a subgroup of G and the inclusion map is not an isomorphism, but it does induce an isomorphism of profinite completions (and related to a famous problem of Grothendieck as in the title we will discuss). We will go onto show how this framework arises naturally in constructing finitely presented groups that are pro finitely rigid amongst finitely presented groups, but not finitely generated ones.

**Last updated:** Sunday, June 23, 2024