Arithmeticity and geometrical commensurators

TL;DR

This paper characterizes rank-one arithmetic and locally symmetric metrics using coarse-geometric commensurators, providing results under the Hilbert-Smith conjecture and for certain negatively curved manifolds.

Contribution

It introduces a coarse-geometric approach to characterize arithmetic and symmetric metrics, extending understanding in geometric group theory.

Findings

Positive characterization under the Hilbert-Smith conjecture

Unconditional results for finite volume negatively curved manifolds

Advances in coarse geometric analysis of symmetric spaces

Abstract

This paper aims to characterize rank-one arithmetic and locally symmetric metrics in the coarsely geometric setting using coarse-geometric commensurators. We provide a positive answer in general under the Hilbert-Smith conjecture and unconditionally for finite volume negatively curved manifolds with finitely many cusps.