Property (RD) and Free Wreath Products

Mardi, 18. octobre 2022 - 14:00 - 15:00

Michael Brannan (Waterloo)


A discrete group $G$ is said to have "the property of rapid decay'' (property (RD) for short) if, roughly speaking, one can control the convolution operator norms of elements of the group algebra $\mathbb C[G]$ in terms of their much easier to compute $\ell^2(G)$-norms.  Property (RD) is a very powerful property of groups, which has broad applications in operator algebras, geometric group theory, and quantum information.  For discrete quantum groups, property (RD) was introduced and studied by Roland Vergnioux. In this talk I'll review property (RD) in the quantum setting, describe some natural examples, and explain a new result which shows that property (RD) is stable with respect to the operation of taking free wreath products of (duals of) discrete groups by quantum automorphism groups of finite dimensioanal tracial C$^\ast$-algebras. This result in particular allows us to see that (the discrete duals of) the hyperoctahedral free quantum groups have property (RD).  (This is joint work with Li Gao and John Weeks).