Property of Rapid Decay and Free Wreath Products

Abstract

Compact quantum groups have been objects of study for nearly fifty years as ``perturbations" of groups, ones whose algebras of ``continuous functions" is defined to be non-commutative. The quantum automorphism group $\mathbb{G}^+(B,\phi)$ is a distinguished example of these as having a neatly-presented representation category yielding several recent results. In particular, the free wreath product construction $\wr_*$ - which mirrors its classical motivation as breaking down the automorphism group of disjoint copies of a graph - has given several results when pairing the quantum automorphism group with a compact quantum group $\widehat{\Gamma}$ for a discrete classical group $\Gamma$. In this work we show that this construction preserves property (RD) wherever it can: that is, under the right circumstances, $\Gamma$ has property (RD) iff $\widehat{\Gamma}\wr_* \mathbb{G}^+(B,\phi)$ has property (RD). We apply the theorems of recent papers on the representation theory of this object together with calculating $\theta$-nets of certain intertwiners to get this result.