Title: Virtual retraction properties in groups

Abstract: In this talk I will give an overview of retractions in groups. Given a group G, we say that a subgroup H is a virtual retract of G when H is a retract of a finite-index subgroup of G. If all cyclic/f.g subgroups of G are virtual retracts, we say that G has (LR)/(VRC).

Subgroups being virtual retracts may tell us if the group is residually finite, LERF, or how do subgroups embed on the whole group. I will discuss some motivations, and then expand on some extension problems: If G has (LR)/(VRC), what about its quotients? If a quotient of G has (LR)/(VRC), what about G? If G has (LR), what about wreath products of G with finite groups?