Benjamini-Schramm and spectral convergence

TL;DR

This paper demonstrates that under mild conditions, Benjamini-Schramm convergence of lattices in locally compact groups is equivalent to spectral convergence, and extends these concepts to the relative case using relative L2-theory.

Contribution

The paper establishes the equivalence of Benjamini-Schramm and spectral convergence under mild conditions and extends these notions to the relative case with a new L2-theory framework.

Findings

Benjamini-Schramm convergence is equivalent to spectral convergence under mild conditions.

Extensions of both notions to the relative case are achieved.

Spectral and Benjamini-Schramm convergence are characterized in terms of relative L2-theory.

Abstract

It is shown that under mild conditions, Benjamini-Schramm convergence of lattices in locally compact groups is equivalent to spectral convergence. Next both notions are extended to the relative case and are then expressed in terms of relative L2-theory.