A note on inner amenability for FLC point sets

TL;DR

This paper proves that groupoids associated with approximate lattices and finite local complexity point sets in second countable locally compact groups are inner amenable, linking groupoid properties with lattice dynamics.

Contribution

It establishes the inner amenability of groupoids related to approximate lattices and finite local complexity point sets, advancing understanding of their operator algebra properties.

Findings

Groupoids from approximate lattices are inner amenable.

Inner amenability extends to point sets with finite local complexity.

Results connect lattice dynamics with operator algebra properties.

Abstract

Inner amenability is a bridge between amenability of an object and amenability of its operator algebras. It is an open problem of Ananantharman-Delaroche to decide whether all \'etale groupoids are inner amenable. Approximate lattices and their dynamics have recently attracted increased attention and have been studied using groupoid methods. In this note, we prove that groupoids associated with approximate lattices in second countable locally compact groups are inner amenable. In fact we show that this result holds more generally for point sets of finite local complexity in such groups.