"Geometry and Analysis on Groups" Research Seminar
Virtual retraction properties in groups.
Ashot Minasyan (Southampton)
A subgroup \(H\) is a retract of a group \(K\) if there is a homomorphism \(\rho:K \to H\) such that the restriction of \(\rho\) to \(H\) is the identity map. This is equivalent to saying that \(K\) decomposes as a semidirect product of a normal subgroup \(N \lhd K\) with \(H\). \(H\) is a virtual retract of a group \(G\) if it is a retract of some finite index subgroup \(K\) of \(G\).
Virtual retracts play an important role in Combinatorial and Geometric Group Theory, and the talk will focus on classes of groups where all finitely generated (respectively, all cyclic) subgroups are virtual retracts. We will look at examples of groups in these two classes, their basic properties and stability under various group-theoretic constructions.