Profinite and residual methods around geometric group theory

In this introductory talk, we will define profinite groups and the profinite completion of a group. We will discuss profinite rigidity and see some examples of profinitely rigid groups. Finally, we will mention some recent exciting results in the area of profinite rigidity.

Homology Growth and Polynomial Mapping Tori

Abstract: (joint work with Yassine Guerch, Sam Hughes and Monika Kudlinska) Given a group, one can consider how the homology groups of its finite index subgroups behave. For instance, you could ask how the torsion part of the homology groups vary (or grow, maybe) as you take deeper and deeper covers. We show that many mapping tori do not exhibit this kind of homology growth: the strategy involves constructing actions on trees and applying a combination theorem due to Abert, Bergeron, Fraczyk, and Gaboriau.

https://arxiv.org/abs/2305.10410

Hyperbolic 3-manifold groups and profinite quotients

Abstract: For an infinite, residually finite group, it is interesting to ask what properties of the group are captured by its finite quotients. We will discuss how to use ideas of Bridson-McReynolds-Reid-Spitler to show, for example, that the fundamental group of zero surgery on the knot 6_2 is completely determined (among all residually finite groups) by the collection of its finite quotients.

https://arxiv.org/abs/2205.08693

Profinite invariants of free-by-cyclic groups

Abstract: A central question in Geometric Group Theory is to determine how much algebraic information is encoded in the set of finite quotients of a given group. A group is said to be (absolutely) profinitely rigid if its isomorphism type is completely determined by its profinite completion. A more tractable problem is to determine profinite rigidity within a certain predetermined class of groups. Much work has been done towards solving this problem for fundamental groups of 3-manifolds. In this 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, and show how such results can be used to obtain relative profinite rigidity. The first part of this talk will focus on the necessary background, as well as some recent work and open problems in the area. No prior knowledge of the topic will be assumed throughout. This is based on joint work with Sam Hughes.

https://arxiv.org/abs/2303.16834 

On quotients of Coxeter groups

Abstract: A group G is said to be just-infinite if G is infinite and every proper quotient of G is finite. This setting appears to be very fruitful for the study of profinite rigidity, i.e. the question to what extend a group is determined by its finite quotients. In this talk I characterize which Coxeter groups are just-infinite and show these are profinitely rigid among the class of Coxeter groups. Furthermore an example of absolute profinite rigidity for Coxeter groups is discussed.

https://arxiv.org/abs/2305.06207

Linear Representations and Finite Quotients 

Abstract: There has recently been much work focused on the problem of distinguishing the fundamental groups of 3-manifolds by using their finite quotients. Many properties of 3-manifolds have been shown to be detectable in this way, and certain 3-manifolds are known to be completely determined by their finite quotients. A few 3-manifold groups are known to be profinitely rigid, they are determined among all finitely generated, residually finite groups by their collection of finite quotients. I will discuss some recent work which describes how the finite quotients of a group can detect its linear representations. Joint work (in preparation) with Ben McReynolds.

Small quotients of surface braid groups

Abstract: The classical braid groups can be viewed from many different angles and admit generalizations in just as many directions. Surface braid groups are a topological generalization of the braid groups that have close connections with mapping class groups of surfaces. In this talk we review a recent result on minimal nonabelian finite quotients of braid groups. In considering the analogous problem for surface braid groups, we construct small nonabelian quotients by generalizing the Heisenberg group which do not arise as quotients of the braid group.

https://arxiv.org/abs/2301.01872