In this talk, I will explain some applications of the characterization of finitely generated subgroups of Thompson group V given in Davide Perego’s talk. I will recall some of the previously known obstructions, start building up by rephrasing them in our new framework, and then give some ideas towards the following stronger alternative: every finitely generated subgroup of V is either virtually abelian, or contains a free non-abelian semigroup.

This is the second part of a two-parts talk with Davide Perego.