Universal systole bounds for arithmetic locally symmetric spaces

TL;DR

This paper establishes universal lower bounds for the systole in certain arithmetic locally symmetric orbifolds, linking geometric properties with number-theoretic bounds and prime splitting behavior.

Contribution

It introduces new bounds for translation lengths in SL_n(R) using Mahler measures and applies geometric methods to number field extensions with prescribed prime splitting.

Findings

Universal systole bounds for arithmetic orbifolds

New translation length bounds in SL_n(R)

Existence of number field extensions with specific prime splitting

Abstract

The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.