Analytic properties of approximate lattices

TL;DR

This paper develops a framework for analyzing the properties of approximate lattices in locally compact groups, linking their Kazhdan- and Haagerup-type properties to those of the ambient groups, and extends to various $L^p$-properties.

Contribution

It introduces cocycle-induction for approximate lattices and relates their properties to the ambient group's properties, covering a broad class of uniform approximate lattices.

Findings

Relates approximate lattice properties to ambient group properties.

Extends analysis to $L^p$-versions of Property (FH) and a-(FH)-menability.

Provides a flexible approach applicable to many uniform approximate lattices.

Abstract

We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the $L^p$-versions of Property (FH) and a-(FH)-menability as well as quasified versions thereof a la Burger--Monod and Ozawa.