On compact bicrossed products

TL;DR

This paper provides a comprehensive study of compact bicrossed products from matched pairs of groups, exploring their properties, constructions, and examples, including new instances of quantum groups with property (T).

Contribution

It introduces an easy construction method for bicrossed products, investigates their properties like property (T) and Haagerup, and identifies new quantum groups with property (T).

Findings

Identified two non-trivial compact bicrossed products with relative co-property (T).

Constructed an infinite family of non-isomorphic discrete quantum groups with property (T).

Established permanence of rapid decay and weak amenability in certain quantum groups.

Abstract

We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and and produce an easy construction of the associated bicrossed product $\mathbb{G}$. We investigate the relative co-property $(T)$ and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property $(T)$ and Haagerup property of the discrete dual $\widehat{\mathbb{G}}$, and review co-amenability of $\mathbb{G}$ as well. We distinguish two such non-trivial compact bicrossed products with relative co-property $(T)$ and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property $(T)$, the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.