Operator algebra seminar

Speaker: Ruben Martos (Paris 7)

Title: Torsion and K-theory for some free wreath products

Abstract: Let $\mathbb{G}$ be a compact quantum group and $N > 4$.

In a first phase, we are going to study the Baum-Connes conjecture for the dual of a free wreath product $\mathbb{G} \wr_{*} S_N^+$ in order to give an appropriate formulation. This brings us to carry out a complete classification of torsion actions of $\mathbb{G} \wr_{*} S_N^+$.

Afterwards, we show that if $\mathbb{G}$ is torsion-free and satisfies the strong Baum-Connes conjecture, then $\widehat{\mathbb{G} \wr_{*} S_N^+}$ satisfies the strong Baum-Connes conjecture.

Finally, this result allows to perform explicit K-theory computations. Especially, we explain the strategy to reach the K-group computations for the $C^*$-algebra defining the free wreath product $\mathbb{G} \wr_* SO_q(3)$, which is monoidally equivalent to $\mathbb{G} \wr_{*} S_N^+$, in three pertinent situations: when $\mathbb{G}$ is an orthogonal quantum group, $\mathbb{G}$ is a free quantum group and $\mathbb{G}$ is a classical free group.

It is a collaboration work with A. Freslon (Université Paris-Sud).