Some Examples of Bicrossed Products with the Rapid Decay Property

TL;DR

This paper investigates bicrossed products of compact semi-direct products with finite subgroups, providing criteria for their duals to have the rapid decay property and constructing examples of quantum groups with this property.

Contribution

It offers a practical criterion for the rapid decay property in bicrossed products and constructs new examples of quantum groups with this property.

Findings

Criteria for rapid decay property in bicrossed products

Examples of quantum groups with rapid decay but non-polynomial growth

Construction of bicrossed products not satisfying the main hypothesis

Abstract

We consider bicrossed products obtained by twisting compact semi-direct products by a suitable finite subgroup. Under some restriction, we give a practical criterion for the discrete dual of such bicrossed products to have the rapid decay property (property (RD)). Using this theory, we construct some examples of discrete quantum groups with (RD) but do not grow polynomially. Further examples that do not satisfy the hypothesis of our main result are also constructed.