Benjamini-Schramm and spectral convergence II. The non-homogeneous case

TL;DR

This paper extends the equivalence between spectral convergence and Benjamini-Schramm convergence from homogeneous spaces to non-homogeneous spaces that are compact modulo isometry group, under a uniform discreteness condition.

Contribution

It generalizes the known equivalence to a broader class of spaces, providing new insights into spectral and geometric convergence in non-homogeneous settings.

Findings

Proves equivalence under uniform discreteness

Extends previous results from homogeneous to non-homogeneous spaces

Identifies open questions regarding implications without the discreteness condition

Abstract

The equivalence of spectral convergence and Benjamini-Schramm convergence is extended from homogeneous spaces to spaces which are compact modulo isometry group. The equivalence is proven under the condition of a uniform discreteness property. It is open, which implications hold without this condition.