On the Whitehead nightmare and some related topics

Abstract

In this very fast survey I will start by reviewing my work on the QSF property in geometric group theory. While working on this topic, I met some infinitistic complications which I called the Whitehead nightmare. This has analogues in other areas of low-dimensional topology. Afterwards I will present some joint work in progress with Louis Funar and Daniele Otera, where we plan to show that all groups, and only finitely presented groups are concerned by this paper, can avoid the nightmare in question. And also that all groups have the GSC (geometric simple connectivity) property, much stronger than QSF. Both QSF and GSC are explained in the main text.