Universal systole bounds for arithmetic locally symmetric spaces

Lapan, Sara; Linowitz, Benjamin; Meyer, Jeffrey

doi:10.1090/proc/15683

Universal systole bounds for arithmetic locally symmetric spaces
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by Sara Lapan, Benjamin Linowitz and Jeffrey S. Meyer PDF

Proc. Amer. Math. Soc. 150 (2022), 795-807 Request permission

Abstract:

The systole of a Riemannian manifold is the minimal length of a non-contractible closed geodesic 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 semisimple $x\in \operatorname {SL}_n(\mathbf {R})$ 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.

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