Geometric property (T) for non-discrete spaces

TL;DR

This paper extends the concept of geometric property (T) to non-discrete coarse spaces, including manifolds, and characterizes it via spectral properties, broadening its applicability beyond discrete settings.

Contribution

It introduces a generalized, coarse-invariant definition of geometric property (T) applicable to non-discrete spaces and links it to spectral properties of Laplacians.

Findings

Broader definition of bounded geometry for coarse spaces.

Geometric property (T) is a coarse invariant.

Characterization of property (T) via Laplacian spectra.

Abstract

Geometric property (T) was defined by Willett and Yu, first for sequences of graphs and later for more general discrete spaces. Increasing sequences of graphs with geometric property (T) are expanders, and they are examples of coarse spaces for which the maximal coarse Baum-Connes assembly map fails to be surjective. Here, we give a broader definition of bounded geometry for coarse spaces, which includes non-discrete spaces. We define a generalisation of geometric property (T) for this class of spaces and show that it is a coarse invariant. Additionally, we characterise it in terms of spectral properties of Laplacians. We investigate geometric property (T) for manifolds and warped systems.