Free wreath products with amalgamation

TL;DR

This paper introduces and analyzes free wreath products with amalgamation for compact quantum groups, extending previous concepts and establishing key properties like residual finiteness and the Haagerup property.

Contribution

It defines free wreath products with amalgamation for quantum groups and provides multiple characterizations, advancing the understanding of their structural properties.

Findings

Proves residual finiteness for these quantum groups

Establishes the Haagerup property for the constructed objects

Provides a smash product decomposition

Abstract

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name "free wreath products of pairs" in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.