Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces

TL;DR

This paper introduces weak commensurability of arithmetic groups and explores its implications for the length spectra and isospectrality of locally symmetric spaces, revealing new connections and consequences.

Contribution

It defines weak commensurability for arithmetic groups and links it to length and spectral equivalences of locally symmetric spaces, providing new theoretical insights.

Findings

Weak commensurability implies length and spectral equivalences.

Many strong consequences of weak commensurability are established.

Results include new insights into isospectral and length-commensurable spaces.

Abstract

We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces.