Amenability of Hecke pairs and a question of Eymard

TL;DR

This paper proves that amenability of a Hecke pair is inherited by subgroups containing the Hecke subgroup, answering a long-standing question of Eymard from 1972.

Contribution

It establishes that if (G,H) is an amenable Hecke pair and Γ contains H, then (Γ,H) is also amenable, confirming a specific inheritance property.

Findings

Amenability is preserved when passing to subgroups containing H.

Answers Eymard's 1972 question in the context of Hecke pairs.

Provides a positive inheritance result for amenability in Hecke pairs.

Abstract

In this short paper we show that if $(G,H)$ is an amenable Hecke pair and $\Gamma$ is a subgroup of $G$ containing $H$, then the Hecke pair $(\Gamma, H)$ is amenable too. This answers positively a question of Pierre Eymard, asked in 1972, when it is restricted to Hecke pairs