Systole inequalities for arithmetic locally symmetric spaces

TL;DR

This paper investigates how the shortest non-contractible loops (systoles) in arithmetic locally symmetric spaces grow with volume, establishing a logarithmic lower bound that extends previous results in hyperbolic geometry.

Contribution

It generalizes prior work by proving logarithmic systole growth for a broader class of arithmetic locally symmetric spaces, beyond hyperbolic manifolds.

Findings

Systole growth is at least logarithmic in volume for these spaces.

Extends previous hyperbolic results to more general locally symmetric spaces.

Provides a unified approach to systole inequalities in arithmetic settings.

Abstract

In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered.