The aim of the present paper is to construct series invariants free knots (flat virtual knots, knots) valued in groups (and also products cyclic groups).

The interpolated free group factors L(Fr) for 1 < r ≤ ∞, (also defined by F. Rădulescu) are given another (but equivalent) definition as well as proofs of their properties with respect to compression by projections and free products. In order to prove the addition formula for free products, algebraic techniques are developed which allow us to show R∗R ∼= L(F2) where R is the hyperfinite II1–fac...

We show the existence of noncommutative random variables with finite free entropy but which do not generate a free group factor. In particular, this gives an example of variables X1, . . . , Xn such that δ(X1, . . . , Xn) = n while W ∗(X1, . . . , Xn) ≇ L(Fn).