Length functions and property (RD) for locally compact Hecke pairs

TL;DR

This paper investigates property (RD) for locally compact Hecke pairs, establishing equivalences with group property (RD), and identifying conditions under which Hecke pairs possess this property, including the role of Schlichting completion.

Contribution

It introduces a framework linking property (RD) of Hecke pairs with that of locally compact groups, enabling transfer of results and identification of new examples.

Findings

Property (RD) of Hecke pairs is equivalent to that of certain groups.

Hecke pairs with property (RD) can be characterized via Schlichting completion.

Relative unimodularity is necessary for discrete Hecke pairs to have property (RD).

Abstract

The purpose of this paper is to study property (RD) for locally compact Hecke pairs. We discuss length functions on Hecke pairs and the growth of Hecke pairs. We establish an equivalence between property (RD) of locally compact groups and property (RD) of certain locally compact Hecke pairs. This allows us to transfer several important results concerning property (RD) of locally compact groups into our setting, and consequently to identify many classes of examples of locally compact Hecke pairs with property (RD). We also show that a reduced discrete Hecke pair $(G,H)$ has (RD) if and only if its Schlichting completion $\bar{G}$ has (RD). Then it follows that the relative unimodularity is a necessary condition for a discrete Hecke pair to possess property (RD).