Weakly commensurable groups, with applications to differential geometry

TL;DR

This paper surveys recent results on weakly commensurable groups, their applications in differential geometry, and related algebraic problems involving maximal tori and division algebras.

Contribution

It provides a comprehensive overview of new findings and conjectures connecting algebraic groups, geometric spaces, and division algebras.

Findings

Classification of weakly commensurable groups and their geometric implications

Results on length-commensurable and isospectral locally symmetric spaces

Conjectures on algebraic groups with identical maximal tori and division algebras

Abstract

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.