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Some Examples of Bicrossed Products with the Rapid Decay Property

Hua Wang

TL;DR

The paper develops practical criteria for when the dual of a bicrossed product has the rapid decay property (RD), deriving both a sufficient condition and a strengthened length-function framework. It analyzes the representation theory of bicrossed products, reduces RD/polynomial-growth questions to compatible length data on the factors, and shows how to construct explicit examples with RD but nonpolynomial growth by twisting semidirect products. Employing results for semidirect products and Jolissaint’s RD for amalgamated products, the work produces new discrete quantum groups with RD, some not growing polynomially, and also highlights natural limitations when the finite-image hypothesis on the induced outer action is dropped. Overall, the study provides a toolkit to build and verify RD instances for a broad class of bicrossed products, connecting length-function compatibility to concrete representation-theoretic data. The findings enhance the catalogue of RD examples in the quantum group setting and clarify when RD does not imply polynomial growth.

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.

Some Examples of Bicrossed Products with the Rapid Decay Property

TL;DR

The paper develops practical criteria for when the dual of a bicrossed product has the rapid decay property (RD), deriving both a sufficient condition and a strengthened length-function framework. It analyzes the representation theory of bicrossed products, reduces RD/polynomial-growth questions to compatible length data on the factors, and shows how to construct explicit examples with RD but nonpolynomial growth by twisting semidirect products. Employing results for semidirect products and Jolissaint’s RD for amalgamated products, the work produces new discrete quantum groups with RD, some not growing polynomially, and also highlights natural limitations when the finite-image hypothesis on the induced outer action is dropped. Overall, the study provides a toolkit to build and verify RD instances for a broad class of bicrossed products, connecting length-function compatibility to concrete representation-theoretic data. The findings enhance the catalogue of RD examples in the quantum group setting and clarify when RD does not imply polynomial growth.

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.
Paper Structure (18 sections, 19 theorems, 82 equations)

This paper contains 18 sections, 19 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Conventions and notations
  3. Acknowledgement
  4. Preliminaries
  5. Rapid decay and polynomial growth for discrete quantum groups of Kac type
  6. Bicrossed products and (RD)
  7. Construction of the bicrossed product
  8. Representation theory of the bicrossed product
  9. Permanence of (RD) and polynomial growth
  10. Representation theory of some semi-direct products
  11. Classification of irreducible representations
  12. The fusion rules
  13. Nontrivial bicrossed product from semidirect product
  14. More notations
  15. Proof of Theorem \ref{['theo:69214233f77b0115']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

In the above settings. If there is a $\Gamma$-invariant length function $l_{\widehat{G}}$ on $\widehat{G}$, and a $\beta^{\Lambda}$-invariant length function $l_{\Gamma}$ on $\Gamma$, such that both $\pair*{\widehat{G}}{l_{\widehat{G}}}$ and $\pair*{\Gamma}{l_{\Gamma}}$ have polynomial growth (resp.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Classification of irreducible representations, phdthesis*p26, Theorem I.4.9 (c)
  • Proposition 2.2
  • Theorem 2.3: Conjugate of irreducible representations, phdthesis*pp.28,29, Definition I.4.12 & Theorem I.4.13
  • Theorem 2.4: Fusion rules
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7: Permanence of polynomial growth
  • Theorem 2.8: Permanence of (RD)
  • ...and 28 more