The Thompson's group F consists of piecewise-linear homeomorphisms of [0,1] with finitely many singularities at dyadic rational points and with slopes equal to integer powers of 2. It is one of the most famous and very important groups related to many areas of mathematics. In this talk, I will review several rigidity type phenomena of F and connections between them:

• absence of non-trivial characters;
• essential freeness of all measure-preserving actions;
• absence of non-trivial invariant random subgroups;
• invariant subalgebras rigidity.

The talk is based on an old joint work with Constantine Medynets and on a recent joint work with Tattwamasi Amrutam.